Matching Items MCQs for Sub-Topics of Topic 10: Calculus

(i) Concept of a Limit

(ii) Left Hand Limit ($\lim\limits_{x \to a^-} f(x)$)

(iii) Right Hand Limit ($\lim\limits_{x \to a^+} f(x)$)

(iv) Existence of a Limit

(v) Indeterminate Form

(a) The function approaches this value as $x$ approaches $a$ from values greater than $a$.

(b) The value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$, but not necessarily equal to $a$.

(c) Occurs when direct substitution yields expressions like $0/0$, $\infty/\infty$, $0 \cdot \infty$, etc.

(d) Requires the left and right hand limits to be equal and finite.

(e) The function approaches this value as $x$ approaches $a$ from values less than $a$.

(i) $\lim\limits_{x \to 2} (x^3 - 5x + 1)$

(ii) $\lim\limits_{x \to 3} \frac{x^2 - 9}{x - 3}$

(iii) $\lim\limits_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$

(iv) $\lim\limits_{x \to 0} \frac{\sin(2x)}{x}$

(v) $\lim\limits_{x \to 1} \frac{x-1}{\sqrt{x^2+3} - 2}$

(a) Rationalization of the denominator

(b) Direct Substitution

(c) Factorization

(d) Using Standard Limit Formulas ($\lim\limits_{x \to 0} \frac{\sin kx}{x} = k$)

(e) Rationalization of the numerator

(i) $\lim\limits_{x \to 1} \frac{x^2 - 1}{x - 1}$

(ii) $\lim\limits_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$

(iii) $\lim\limits_{x \to 0} \frac{|x|}{x}$

(iv) $\lim\limits_{x \to 2} (x+5)$

(v) $\lim\limits_{x \to 0} \frac{\sin(4x)}{x}$

(a) 7

(b) Does not exist

(c) 1/2

(d) 2

(e) 4

(i) $f(x) = \begin{cases} x^2+1 & , & x \leq 0 \\ x-1 & , & x > 0 \end{cases}$

(ii) $f(x) = |x|$

(iii) $f(x) = \frac{|x|}{x}$

(iv) $f(x) = \lfloor x \rfloor$

(v) $f(x) = \sin x$

(a) 0

(b) 1

(c) -1

(d) Does not exist

(e) -1 (for $\lfloor x \rfloor$ as $x \to 0^-$)

(i) $f(x) = \begin{cases} x+1 & , & x < 1 \\ 2-x & , & x \geq 1 \end{cases}$

(ii) $f(x) = \frac{1}{x-1}$

(iii) $f(x) = \lfloor x \rfloor$

(iv) $f(x) = \frac{x^2-1}{x-1}$

(v) $f(x) = |x-1|$

(a) 0

(b) 1

(c) 2

(d) $\infty$

(e) Undefined

(i) Sum Rule

(ii) Product Rule

(iii) Quotient Rule

(iv) Constant Multiple Rule

(v) Power Rule for Limits ($\lim\limits_{x \to a} (f(x))^n$)

(a) $c L$

(b) $L^n$

(c) $L/M$ (if $M \neq 0$)

(d) $L+M$

(e) $L \cdot M$

(i) $\lim\limits_{x \to 0} \frac{\sin x}{x}$

(ii) $\lim\limits_{x \to 0} \frac{e^x - 1}{x}$

(iii) $\lim\limits_{x \to 0} \frac{\log_e(1+x)}{x}$

(iv) $\lim\limits_{x \to \infty} (1 + \frac{1}{x})^x$

(v) $\lim\limits_{x \to 0} \frac{a^x - 1}{x}$ ($a>0$)

(a) $e$

(b) 1

(c) $\log_e a$

(d) 1

(e) 1

(i) $\lim\limits_{x \to 0} \frac{\tan x}{x}$

(ii) $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2}$

(iii) $\lim\limits_{x \to 0} x \cdot \text{cosec } x$

(iv) $\lim\limits_{x \to 0} \frac{\sin ax}{\sin bx}$ ($a, b \neq 0$)

(v) $\lim\limits_{x \to 0} \frac{1 - \cos(2x)}{x^2}$

(a) 1

(b) a/b

(c) 1/2

(d) 1

(e) 2

(i) $\lim\limits_{x \to a} \frac{x^n - a^n}{x - a}$

(ii) $\lim\limits_{x \to 0} \frac{\sin(kx)}{x}$

(iii) $\lim\limits_{x \to 0} \frac{e^{kx} - 1}{x}$

(iv) $\lim\limits_{x \to 0} \frac{\log_e(1+kx)}{x}$

(v) $\lim\limits_{x \to \infty} (1 + \frac{k}{x})^x$

(a) $e^k$

(b) $n a^{n-1}$

(c) $k$

(d) $k$

(e) $k$

(i) Guarantees the limit of a function is squeezed between two other functions with the same limit.

(ii) States that if a function is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, there is $c \in (a,b)$ where $f'(c)=0$.

(iii) States that if a function is continuous on $[a,b]$ and differentiable on $(a,b)$, there is $c \in (a,b)$ where $f'(c) = \frac{f(b)-f(a)}{b-a}$.

(iv) Relates the derivative of an integral with a variable upper limit to the integrand.

(v) Provides a method to evaluate definite integrals using antiderivatives.

(a) Fundamental Theorem of Calculus (Part 2)

(b) Rolle's Theorem

(c) Fundamental Theorem of Calculus (Part 1)

(d) Squeeze Play Theorem (Sandwich Theorem)

(e) Lagrange's Mean Value Theorem

(i) Removable Discontinuity

(ii) Jump Discontinuity

(iii) Infinite Discontinuity

(iv) Continuity at a point

(v) Continuity on an open interval

(a) The left and right hand limits exist but are not equal.

(b) The limit exists, but the function value is different or undefined at the point.

(c) The limit is infinite at the point.

(d) The limit exists, the function value is defined, and they are equal.

(e) The function is continuous at every point in the interval.

(i) $f(x) = x^2 + \sin x$

(ii) $f(x) = |x|$

(iii) $f(x) = \frac{1}{x}$

(iv) $f(x) = \frac{\sin x}{x}$ (with $f(0)=1$)

(v) $f(x) = \lfloor x \rfloor$

(a) Continuous

(b) Continuous

(c) Infinite discontinuity

(d) Removable discontinuity (if $f(0)$ were undefined)

(e) Jump discontinuity

(i) $f(g(x))$ where $g$ is continuous at $a$ and $f$ is continuous at $g(a)$

(ii) $|f(x)|$ where $f$ is continuous at $a$

(iii) $f(x) + g(x)$ where $f, g$ are continuous at $a$

(iv) $f(x)/g(x)$ where $f, g$ are continuous at $a$

(v) $\sqrt{f(x)}$ where $f$ is continuous at $a$ and $f(a) \geq 0$

(a) Continuous at $a$ if $g(a) \neq 0$

(b) Continuous at $a$

(c) Continuous at $a$

(d) Continuous at $a$

(e) Continuous at $a$

(i) Continuity on an open interval $(a, b)$

(ii) Continuity on a closed interval $[a, b]$

(iii) Right continuity at $a$

(iv) Left continuity at $b$ ($b>a$)

(v) Continuity at a point $c$ in $(a,b)$

(a) $\lim\limits_{x \to c} f(x) = f(c)$

(b) Continuous at every point in $(a, b)$

(c) Continuous on $(a, b)$ plus right continuity at $a$ and left continuity at $b$.

(d) $\lim\limits_{x \to a^+} f(x) = f(a)$

(e) $\lim\limits_{x \to b^-} f(x) = f(b)$

(i) $f(x) = \frac{x^2-4}{x-2}$ at $x=2$

(ii) $f(x) = \begin{cases} x & , & x \neq 1 \\ 5 & , & x = 1 \end{cases}$ at $x=1$ ($\lim\limits_{x \to 1} f(x) = 1 \neq f(1)$)

(iii) $f(x) = \frac{1}{x-3}$ at $x=3$

(iv) $f(x) = \frac{|x|}{x}$ at $x=0$

(v) $f(x) = \tan x$ at $x=\pi/2$

(a) Jump Discontinuity

(b) Removable Discontinuity

(c) Infinite Discontinuity

(d) Removable Discontinuity (original function without redefined point)

(e) Infinite Discontinuity

(i) Derivative $f'(a)$

(ii) Left Hand Derivative ($f'(a^-)$)

(iii) Right Hand Derivative ($f'(a^+)$)

(iv) Definition of derivative at $x=a$ using $x \to a$

(v) Differentiability at $x=a$

(a) $\lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a}$

(b) $\lim\limits_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$

(c) Requires $f'(a^-) = f'(a^+) =$ finite value.

(d) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$

(e) $\lim\limits_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$

(i) $f(x) = x^2$

(ii) $f(x) = |x|$

(iii) $f(x) = \lfloor x \rfloor$

(iv) $f(x) = \begin{cases} x \sin(1/x) & , & x \neq 0 \\ 0 & , & x = 0 \end{cases}$

(v) $f(x) = x|x|$

(a) Differentiable

(b) Not differentiable (continuous, sharp corner)

(c) Not differentiable (discontinuous)

(d) Not differentiable (continuous, oscillating slope)

(e) Differentiable

(i) If a function is differentiable at a point, it is continuous at that point.

(ii) If a function is continuous at a point, it is differentiable at that point.

(iii) If a function is not continuous at a point, it is not differentiable at that point.

(iv) A function can have a vertical tangent at a point where it is continuous but not differentiable.

(v) The derivative is defined as the limit of the difference quotient.

(a) True

(b) False

(c) True

(d) True

(e) True

(i) $f(x) = c$

(ii) $f(x) = x$}

(iii) $f(x) = x^2$

(iv) $f(x) = x^3$

(v) $f(x) = \sin x$

(a) $\cos a$

(b) 0

(c) $2a$}

(d) 1

(e) $3a^2$}

(i) $f'(a)$

(ii) $f'(a) > 0$

(iii) $f'(a) < 0$

(iv) $f'(a) = 0$}

(v) $f'(a)$ is undefined (and $f$ is continuous at $a$)

(a) Tangent is horizontal

(b) Slope of the tangent at $(a, f(a))$

(c) Function is decreasing at $a$

(d) Sharp corner or cusp at $(a, f(a))$

(e) Function is increasing at $a$

(i) Power Rule

(ii) Product Rule

(iii) Quotient Rule

(iv) Sum Rule

(v) Constant Multiple Rule

(a) $(uv)' = u'v + uv'$

(b) $(u/v)' = (u'v - uv')/v^2$

(c) $(u+v)' = u' + v'$

(d) $(cu)' = c u'$

(e) $(x^n)' = nx^{n-1}$}

(i) $\sin x$

(ii) $\cos x$

(iii) $\tan x$

(iv) $e^x$

(v) $\log_e x$

(a) $1/x$

(b) $\cos x$

(c) $-\sin x$

(d) $\sec^2 x$

(e) $e^x$

(i) $\cot x$

(ii) $\sec x$

(iii) $\text{cosec } x$

(iv) $a^x$ ($a>0$)

(v) $\log_a x$ ($a>0, a \neq 1$)

(a) $a^x \log_e a$

(b) $\sec x \tan x$

(c) $- \text{cosec } x \cot x$

(d) $- \text{cosec}^2 x$

(e) $1/(x \log_e a)$

(i) $x^2 \sin x$

(ii) $(x+1)/x^2$

(iii) $\cos x + e^x$

(iv) $5x^{10}$}

(v) $\sqrt{x}$

(a) $-\sin x + e^x$

(b) $50x^9$

(c) $1/(2\sqrt{x})$

(d) $2x \sin x + x^2 \cos x$

(e) $- (x+2)/x^3$}

(i) $\frac{d}{dx}(f(x) + g(x))$

(ii) $\frac{d}{dx}(f(x)g(x))$

(iii) $\frac{d}{dx}(\frac{f(x)}{g(x)})$

(iv) $\frac{d}{dx}(x^n)$

(v) $\frac{d}{dx}(c)$

(a) Quotient Rule

(b) Sum Rule

(c) Constant Rule

(d) Power Rule

(e) Product Rule

(i) $\sin(x^2)$

(ii) $(2x+1)^5$

(iii) $e^{\cos x}$

(iv) $\log_e(x^3+2)$

(v) $\sqrt{\tan x}$

(a) $f(u) = \sqrt{u}$, $u = \tan x$

(b) $f(u) = \sin u$, $u = x^2$

(c) $f(u) = u^5$, $u = 2x+1$

(d) $f(u) = e^u$, $u = \cos x$

(e) $f(u) = \log_e u$, $u = x^3+2$

(i) $\sin(3x)$

(ii) $(x^2+1)^3$}

(iii) $e^{2x}$

(iv) $\log_e(5x)$

(v) $\cos(x^4)$

(a) $3 \sin^2(x^2+1) \cdot 2x$

(b) $3 \cos(3x)$

(c) $3(x^2+1)^2 \cdot 2x = 6x(x^2+1)^2$

(d) $1/x$}

(e) $2e^{2x}$

(i) $y = f(g(x))$

(ii) $y = \sin(g(x))$

(iii) $y = e^{g(x)}$

(iv) $y = (g(x))^n$}

(v) $y = \log_e(g(x))$

(a) $\frac{1}{g(x)} \cdot g'(x)$

(b) $f'(g(x)) \cdot g'(x)$

(c) $e^{g(x)} \cdot g'(x)$

(d) $\cos(g(x)) \cdot g'(x)$

(e) $n(g(x))^{n-1} \cdot g'(x)$}

(i) Rate of change of Revenue w.r.t. Time

(ii) Marginal Revenue

(iii) Rate of change of Sales w.r.t. Time

(iv) Rate of change of Cost w.r.t. Time (C=Cost, x=Units, t=Time)

(v) Marginal Cost

(a) $R'(x)$

(b) $x'(t)$

(c) $C'(x)$

(d) $\frac{dR}{dx} \cdot \frac{dx}{dt}$

(e) $\frac{dC}{dx} \cdot \frac{dx}{dt}$

(i) $\sin^2 x$

(ii) $\cos(2x)$

(iii) $\log_e(\tan x)$

(iv) $\sqrt{e^x}$

(v) $\tan^{-1}(x/2)$

(a) $e^x / (2\sqrt{e^x}) = \sqrt{e^x}/2$}

(b) $2 \sin x \cos x = \sin(2x)$

(c) $\frac{1}{1+(x/2)^2} \cdot (1/2) = \frac{1}{2(1+x^2/4)} = \frac{2}{4+x^2}$}

(d) $\cot x$}

(e) $-2 \sin(2x)$

(i) $x^2 + y^2 = c^2$

(ii) $xy = c$

(iii) $x^3 + y^3 = 3xy$

(iv) $\sin y = x$

(v) $e^{xy} = c$

(a) $-y/x$

(b) $1/\cos y$}

(c) $\frac{y-x^2}{y^2-x}$

(d) $-x/y$

(e) $-y/x$

(i) $\sin^{-1} x$

(ii) $\cos^{-1} x$

(iii) $\tan^{-1} x$

(iv) $\cot^{-1} x$

(v) $\sec^{-1} x$

(a) $-1/\sqrt{1-x^2}$

(b) $1/(1+x^2)$

(c) $-1/(1+x^2)$

(d) $1/(|x|\sqrt{x^2-1})$

(e) $1/\sqrt{1-x^2}$}

(i) $\frac{dy}{dx}$

(ii) $f'(y)$

(iii) Derivative of $\sin^{-1} x$ at $x=1/2$

(iv) Derivative of $\tan^{-1} x$ at $x=1$

(v) Derivative of $\cos^{-1} x$ at $x=0$

(a) $1/f'(y)$

(b) $1/\frac{dx}{dy}$

(c) $1/\sqrt{1-(1/2)^2} = 1/\sqrt{3/4} = 2/\sqrt{3}$}

(d) $-1/\sqrt{1-0^2} = -1$}

(e) $1/(1+1^2) = 1/2$}

(i) $\sin^{-1}(2x\sqrt{1-x^2})$

(ii) $\cos^{-1}(\frac{1-x^2}{1+x^2})$

(iii) $\tan^{-1}(\frac{2x}{1-x^2})$

(iv) $\sin^{-1} x + \cos^{-1} x$

(v) $\tan^{-1}(\frac{3x-x^3}{1-3x^2})$

(a) $3 \tan^{-1} x$

(b) $2 \sin^{-1} x$

(c) $2 \tan^{-1} x$

(d) $\pi/2$

(e) $2 \tan^{-1} x$

(i) $\sin^{-1}(2x)

(ii) $\cos^{-1}(x^2)

(iii) $\tan^{-1}(\sqrt{x})$

(iv) $\cot^{-1}(1/x)$

(v) $\sec^{-1}(e^x)

(a) $\frac{1}{1+(\sqrt{x})^2} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{2\sqrt{x}(1+x)}$

(b) $\frac{1}{\sqrt{1-(2x)^2}} \cdot 2 = \frac{2}{\sqrt{1-4x^2}}$

(c) $\frac{-1}{\sqrt{1-(x^2)^2}} \cdot 2x = \frac{-2x}{\sqrt{1-x^4}}$

(d) $\frac{1}{|e^x|\sqrt{(e^x)^2-1}} \cdot e^x = \frac{e^x}{e^x\sqrt{e^{2x}-1}} = \frac{1}{\sqrt{e^{2x}-1}}$

(e) $\frac{-1}{1+(1/x)^2} \cdot (-1/x^2) = \frac{1}{x^2(1+1/x^2)} = \frac{1}{x^2+1}$}

(i) $y = f(x)^{g(x)}$

(ii) $y = (f(x)g(x)h(x)) / (k(x)l(x))$

(iii) $y = \sin(x^2)$

(iv) $x = f(t), y = g(t)$

(v) $x^2+y^2 = c$

(a) Chain Rule

(b) Logarithmic Differentiation

(c) Parametric Differentiation

(d) Implicit Differentiation

(e) Logarithmic Differentiation

(i) $y = x^x$

(ii) $y = (\sin x)^x$

(iii) $y = x^{\sin x}$

(iv) $y = (\log_e x)^x$

(v) $y = x^{\log_e x}$

(a) $x^{\sin x} (\cos x \log_e x + \sin x/x)$

(b) $x^x (1 + \log_e x)$

(c) $(\sin x)^x (\log_e \sin x + x \cot x)$

(d) $(\log_e x)^x (1/\log_e x + \log_e(\log_e x))$

(e) $x^{\log_e x} (2 \log_e x / x)$

(i) $x = f(t), y = g(t)$

(ii) Find $\frac{dx}{dt}$

(iii) Find $\frac{dy}{dt}$

(iv) Calculate $\frac{dy}{dx}$ formula

(v) Calculate $\frac{d^2 y}{dx^2}$ formula

(a) $f'(t)$

(b) $\frac{dy/dt}{dx/dt}$

(c) $\frac{d}{dt}(\frac{dy}{dx}) / \frac{dx}{dt}$

(d) $g'(t)$

(e) Given definition

(i) $x = t^2, y = t^3$

(ii) $x = \cos \theta, y = \sin \theta$

(iii) $x = at, y = a/t$

(iv) $x = e^t, y = e^{-t}$

(v) $x = a \cos \theta, y = b \sin \theta$

(a) $-\frac{b}{a} \cot \theta$

(b) $\frac{3}{2} t$

(c) $-1/t^2$

(d) $-\cot \theta$

(e) $-e^{-t}/e^t = -e^{-2t}$}

(i) $y = x^5 + \sin x$

(ii) $y = \sin(x^5)$

(iii) $y = x^{\sin 5}$

(iv) $y = x^{\sin x}$

(v) $y = \frac{\sin x \cos x}{\sqrt{x^2+1}}$

(a) Standard Rules (Sum/Power/Trig)

(b) Logarithmic Differentiation

(c) Chain Rule

(d) Logarithmic Differentiation (or Quotient/Product/Chain)

(e) Standard Rule (Power)

(i) $x^3$

(ii) $\sin x$

(iii) $\cos x$

(iv) $e^x$

(v) $\log_e x$

(a) $-\sin x$

(b) $6x$

(c) $-1/x^2$

(d) $-\cos x$

(e) $e^x$

(i) $x^4$

(ii) $\sin x$

(iii) $e^{2x}$

(iv) $\log_e x$}

(v) $x^2$

(a) 0

(b) $-\cos x$

(c) $6x$}

(d) $8e^{2x}$

(e) $2/x^3$}

(i) $\frac{d^2 y}{dx^2}$

(ii) Point of inflection

(iii) Concave upwards

(iv) Concave downwards

(v) Derivative of a polynomial of degree $n$ ($n$-th derivative)

(a) Where concavity changes (often $y''=0$ or undefined)

(b) $y'' < 0$

(c) Second derivative

(d) $y'' > 0$

(e) A constant

(i) $y = e^{ax}$

(ii) $y = x^m$ ($n \leq m$ integer)

(iii) $y = \sin(ax+b)$

(iv) $y = \cos(ax+b)$

(v) $y = \frac{1}{x}$}

(a) $a^n e^{ax}$

(b) $a^n \sin(ax+b + n\pi/2)$}

(c) $m!/(m-n)! x^{m-n}$}

(d) $(-1)^n n! / x^{n+1}$}

(e) $a^n \cos(ax+b + n\pi/2)$}

(i) $x \sin x$

(ii) $x e^x$

(iii) $x \log_e x$

(iv) $\sin(x^2)$

(v) $(x+1)^3$}

(a) $e^x(x+2)$}

(b) $6(x+1)$}

(c) $\cos(x^2) - 4x^2 \sin(x^2)$}

(d) $1/x$}

(e) $2 \cos x - x \sin x$}

(i) $f(a) = f(b)$

(ii) $f'(c) = 0$

(iii) $f'(c) = (f(b)-f(a))/(b-a)$

(iv) Applies if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$.

(v) Applies if $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a)=f(b)$.

(a) Condition for Rolle's Theorem

(b) Conclusion of Rolle's Theorem

(c) Conclusion of Lagrange's Mean Value Theorem

(d) Conditions for Lagrange's Mean Value Theorem

(e) Conditions for Rolle's Theorem

(i) Tangent is horizontal somewhere between $a$ and $b$ if $f(a)=f(b)$.

(ii) Tangent is parallel to the chord joining $(a, f(a))$ and $(b, f(b))$ somewhere between $a$ and $b$.

(iii) The instantaneous rate of change equals the average rate of change over the interval $[a, b]$.

(iv) If $f(a)=f(b)$, the slope of the tangent at some point is zero.

(v) The slope of the tangent at some point equals the slope of the secant line connecting the endpoints.

(a) Geometric interpretation of Rolle's Theorem

(b) Geometric interpretation of Lagrange's Mean Value Theorem

(c) Interpretation of $f'(c) = \frac{f(b)-f(a)}{b-a}$

(d) Geometric interpretation of Rolle's Theorem

(e) Geometric interpretation of Lagrange's Mean Value Theorem

(i) $f(x) = x^2-4$ on $[-2, 2]$

(ii) $f(x) = |x|$ on $[-1, 1]$

(iii) $f(x) = \frac{1}{x}$ on $[1, 2]$

(iv) $f(x) = \sin x$ on $[0, \pi]$

(v) $f(x) = \tan x$ on $[0, \pi]$

(a) Not applicable (not continuous)

(b) Applicable

(c) Not applicable (not differentiable in interval)

(d) Not applicable (f(a) != f(b))

(e) Applicable

(i) $f(x) = x^2 - 2x$ on $[0, 2]$

(ii) $f(x) = \sin x$ on $[0, \pi]$

(iii) $f(x) = x^2 - 4x + 3$ on $[1, 3]$

(iv) $f(x) = \cos x$ on $[0, 2\pi]$

(v) $f(x) = (x-1)(x-2)$ on $[1, 2]$

(a) $\pi/2, 3\pi/2$

(b) $1.5$ (midpoint)

(c) 1

(d) 2

(e) $\pi/2$

(i) $f(x) = x^2$ on $[0, 2]$

(ii) $f(x) = \log_e x$ on $[1, e]$

(iii) $f(x) = e^x$ on $[0, 1]$

(iv) $f(x) = x^3$ on $[1, 2]$

(v) $f(x) = x^2+2x$ on $[0, 1]$

(a) $2c = (2^2-0^2)/(2-0) = 4/2=2$, $c=1$. (Matches (i))

(b) $1/c = (\log_e e - \log_e 1)/(e-1) = (1-0)/(e-1) = 1/(e-1)$, $c=e-1$. (Matches (ii))

(c) $e^c = (e^1-e^0)/(1-0) = (e-1)/1 = e-1$, $c=\log_e(e-1)$. (Matches (iii))

(d) $3c^2 = (2^3-1^3)/(2-1) = (8-1)/1 = 7$, $c^2 = 7/3$, $c=\sqrt{7/3}$. (Matches (iv))

(e) $f'(x) = 2x+2$. $f'(c) = 2c+2 = (f(1)-f(0))/(1-0) = ((1^2+2(1)) - (0^2+2(0)))/1 = (3-0)/1 = 3$. $2c+2=3$, $2c=1$, $c=1/2$. (Matches (v))

(i) Area of a circle $A = \pi r^2$

(ii) Volume of a sphere $V = \frac{4}{3}\pi r^3$

(iii) Surface area of a cube $S = 6 s^2$

(iv) Volume of a cube $V = s^3$

(v) Perimeter of a square $P = 4s$

(a) $\frac{dV}{dt} = 3s^2 \frac{ds}{dt}$

(b) $\frac{dP}{dt} = 4 \frac{ds}{dt}$

(c) $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

(d) $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$

(e) $\frac{dS}{dt} = 12s \frac{ds}{dt}$

(i) Marginal Cost

(ii) Marginal Revenue

(iii) Marginal Profit

(iv) Rate of change of Cost w.r.t. Units

(v) Rate of change of Revenue w.r.t. Units

(a) $R'(x)$

(b) $C'(x)$

(c) $P'(x)$

(d) $C'(x)$

(e) $R'(x)$

(i) $C(x) = 500 + 10x$

(ii) $C(x) = 1000 + 5x + 2x^2$

(iii) $C(x) = x^3 - 3x + 100$

(iv) $C(x) = 25x$

(v) $C(x) = 0.01x^2 + 2x + 50$

(a) $0.02x + 2$

(b) $3x^2 - 3$

(c) $5 + 4x$

(d) 10

(e) 25

(i) $R(x) = 100x$

(ii) $R(x) = 10x - x^2$

(iii) $R(x) = 5x^2 + 20x$

(iv) $R(x) = x(100 - 0.5x) = 100x - 0.5x^2$

(v) $R(x) = 100 + 50x - x^2/4$

(a) $100 - x/2$

(b) $10 - 2x$

(c) 100

(d) $10x + 20$

(e) $50 - x/2$}

(i) Water flowing into a cylindrical tank (Volume V, radius r, height h)

(ii) Expanding circle (Area A, radius r)

(iii) Deflating spherical balloon (Volume V, radius r)

(iv) Ladder sliding down a wall (x=dist from wall, y=height on wall)

(v) Growing equilateral triangle (Area A, side s)

(a) $A = \pi r^2$, $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

(b) $V = \frac{4}{3}\pi r^3$, $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$

(c) $x^2 + y^2 = L^2$, $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$

(d) $V = \pi r^2 h$, $\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$ (if r is constant)

(e) $A = \frac{\sqrt{3}}{4} s^2$, $\frac{dA}{dt} = \frac{\sqrt{3}}{2} s \frac{ds}{dt}$

(i) Slope of Tangent at $x_0$

(ii) Equation of Tangent at $(x_0, y_0)$

(iii) Slope of Normal at $x_0$ (if tangent slope is $m_T \neq 0$)

(iv) Equation of Normal at $(x_0, y_0)$ (if tangent slope is $m_T \neq 0$)

(v) Approximate change in $y=f(x)$ ($\Delta y$) for small $\Delta x$

(a) $y - y_0 = (-1/m_T)(x - x_0)$

(b) $f'(x_0)$

(c) $y - y_0 = m_T(x - x_0)$

(d) $-1/f'(x_0)$

(e) $dy = f'(x) \Delta x$

(i) $y = x^2$ at $x=2$

(ii) $y = \sin x$ at $x=0$

(iii) $y = e^x$ at $x=1$

(iv) $y = \log_e x$ at $x=1$

(v) $y = x^3 - x$ at $x=1$

(a) $e$

(b) 1

(c) $3(1)^2 - 1 = 2$

(d) 4

(e) 1

(i) Absolute Error in $y=f(x)$

(ii) Relative Error in $y=f(x)$

(iii) Percentage Error in $y=f(x)$

(iv) Measured quantity

(v) Error in measurement

(a) $\Delta x$ (or $dx$)

(b) $|\Delta y|$ (or $|dy| = |f'(x) \Delta x|$) for a small $\Delta x$

(c) $x$

(d) $\frac{|\Delta y|}{y}$ (or $\frac{|dy|}{y}$) for a small $\Delta x$

(e) $\frac{|\Delta y|}{y} \times 100\%$ (or $\frac{|dy|}{y} \times 100\%$) for a small $\Delta x$

(i) Area of square $A=x^2$

(ii) Volume of cube $V=x^3$}

(iii) Area of circle $A=\pi x^2$

(iv) Perimeter of square $P=4x$

(v) Volume of sphere $V=\frac{4}{3}\pi x^3$

(a) $dP = 4 \Delta x$

(b) $dA = 2\pi x \Delta x$

(c) $dA = 2x \Delta x$

(d) $dV = 3x^2 \Delta x$

(e) $dV = 4\pi x^2 \Delta x$

(i) % Error in side of square (s) -> % Error in Area (s$^2$)

(ii) % Error in radius of circle (r) -> % Error in Area ($\pi$r$^2$)

(iii) % Error in edge of cube (s) -> % Error in Volume (s$^3$)

(iv) % Error in radius of sphere (r) -> % Error in Volume ($\frac{4}{3}\pi$r$^3$)

(v) % Error in length of pendulum (L) -> % Error in Time Period ($T \propto \sqrt{L}$)

(a) $2 \times (\% \text{ Error in r})$

(b) $1/2 \times (\% \text{ Error in L})$

(c) $3 \times (\% \text{ Error in s})$

(d) $2 \times (\% \text{ Error in s})$

(e) $3 \times (\% \text{ Error in r})$

(i) For $x_1 < x_2$, $f(x_1) < f(x_2)$

(ii) For $x_1 < x_2$, $f(x_1) \leq f(x_2)$

(iii) For $x_1 < x_2$, $f(x_1) > f(x_2)$

(iv) For $x_1 < x_2$, $f(x_1) \geq f(x_2)$

(v) A function that is either increasing or decreasing (or constant) on an interval

(a) Strictly Decreasing

(b) Increasing (Non-decreasing)

(c) Monotonic

(d) Strictly Increasing

(e) Decreasing (Non-increasing)

(i) $f'(x) > 0$

(ii) $f'(x) \geq 0$

(iii) $f'(x) < 0$

(iv) $f'(x) \leq 0$

(v) $f'(x) = 0$

(a) Decreasing

(b) Strictly Increasing

(c) Constant

(d) Strictly Decreasing

(e) Increasing

(i) $e^x$

(ii) $x^2$

(iii) $\log_e x$

(iv) $\sin x$ in $[0, \pi]$

(v) $\tan x$ in $(-\pi/2, \pi/2)$

(a) $(0, \infty)$

(b) $(0, \pi/2)$

(c) $(-\infty, \infty)$

(d) $(-\pi/2, \pi/2)$

(e) $(0, \infty)$

(i) $e^{-x}$

(ii) $-x^3$

(iii) $\cos x$ in $[0, \pi]$

(iv) $\cot^{-1} x$

(v) $x^2$

(a) $(-\infty, 0)$

(b) $(0, \pi)$

(c) $(-\infty, \infty)$

(d) $(-\infty, \infty)$

(e) $[0, \pi]$

(i) $f'(x) = 3x^2-3$

(ii) Critical points

(iii) Interval where $f'(x) > 0$

(iv) Interval where $f'(x) < 0$

(v) Interval of strict increase

(a) $(-1, 1)$

(b) $x=\pm 1$

(c) $(-\infty, -1) \cup (1, \infty)$

(d) $(-\infty, -1) \cup (1, \infty)$

(e) Derivative of $f(x)$

(i) Local Maximum

(ii) Local Minimum

(iii) Critical Point

(iv) Absolute Maximum

(v) Absolute Minimum

(a) The smallest value of the function on its entire domain or specified interval.

(b) Where $f'(c)=0$ or $f'(c)$ is undefined (and $c$ is in domain).

(c) The largest value of the function on its entire domain or specified interval.

(d) $f(c)$ is the largest value in a small neighborhood around $c$.

(e) $f(c)$ is the smallest value in a small neighborhood around $c$.

(i) $f'(x)$ changes from + to - as $x$ increases through $c$

(ii) $f'(x)$ changes from - to + as $x$ increases through $c$

(iii) $f'(x)$ does not change sign as $x$ increases through $c$

(iv) $f(c)$ is defined and $f'(x)$ changes sign at $c$

(v) $f(c)$ is defined and $f'(x)$ does not change sign at $c$

(a) No local extremum at $c$ (possibly inflection point)

(b) Local Minimum at $c$

(c) Local Extremum at $c$ (Local Max or Min)

(d) Local Maximum at $c$

(e) No local extremum at $c$ (possibly inflection point)

(i) $f''(c) > 0$

(ii) $f''(c) < 0$

(iii) $f''(c) = 0$

(iv) The function is concave up at $c$

(v) The function is concave down at $c$

(a) Local Maximum at $c$

(b) Test is inconclusive

(c) Local Minimum at $c$ (from Second Derivative Test)

(d) Corresponds to $f''(c) < 0$

(e) Corresponds to $f''(c) > 0$

(i) $x^2 - 4x + 5$

(ii) $x^3 - 3x + 2$

(iii) $|x|$

(iv) $x^3$}

(v) $x^4$}

(a) $x=\pm 1$

(b) $x=0$ (derivative undefined)

(c) $x=0$ (derivative is zero)

(d) $x=2$

(e) $x=0$ (derivative is zero)

(i) Find two numbers with fixed sum, max product.

(ii) Find dimensions of rectangle with fixed perimeter, max area.

(iii) Find dimensions of box with fixed volume, min surface area.

(iv) Minimize cost function $C(x)$.

(v) Maximize profit function $P(x) = R(x) - C(x)$.

(a) Area of the rectangle

(b) Product of the two numbers

(c) Profit function $P(x)$

(d) Surface area of the box

(e) Cost function $C(x)$

(i) Antiderivative of $f(x)$

(ii) Indefinite Integral of $f(x)$

(iii) Constant of Integration

(iv) Integrand

(v) Integration symbol

(a) $\int$

(b) The function being integrated, $f(x)$ in $\int f(x) dx$

(c) A function $F(x)$ such that $F'(x) = f(x)$

(d) The family of all antiderivatives, $F(x) + C$

(e) The arbitrary constant $C$ in the indefinite integral

(i) $x^n$ ($n \neq -1$)

(ii) $1/x$

(iii) $\sin x$

(iv) $\cos x$

(v) $e^x$

(a) $e^x + C$

(b) $\log_e |x| + C$

(c) $\frac{x^{n+1}}{n+1} + C$

(d) $\sin x + C$

(e) $-\cos x + C$

(i) $\sec^2 x$

(ii) $\text{cosec}^2 x$

(iii) $\sec x \tan x$

(iv) $\text{cosec } x \cot x$

(v) $a^x$ ($a>0, a \neq 1$)

(a) $\sec x + C$

(b) $\tan x + C$

(c) $-\cot x + C$

(d) $\frac{a^x}{\log_e a} + C$

(e) $-\text{cosec } x + C$

(i) Differentiation

(ii) Finding an Antiderivative

(iii) Integrating $f(x)$

(iv) Differentiating $F(x)+C$

(v) Differentiating $\int f(x) dx$

(a) $f(x)$

(b) Integration (finding antiderivative)

(c) $f'(x)$

(d) Differentiation

(e) Antidifferentiation

(i) Marginal Cost $C'(x)$

(ii) Marginal Revenue $R'(x)$

(iii) Rate of population growth $P'(t)$

(iv) Velocity $v(t)$ (rate of change of position $s(t)$)

(v) Acceleration $a(t)$ (rate of change of velocity $v(t)$)

(a) Integrate w.r.t. $x$ to find Total Cost $C(x)$

(b) Integrate w.r.t. $x$ to find Total Revenue $R(x)$

(c) Integrate w.r.t. $t$ to find Population $P(t)$

(d) Integrate w.r.t. $t$ to find Position $s(t)$

(e) Integrate w.r.t. $t$ to find Velocity $v(t)$

(i) $\int x \cos x dx$

(ii) $\int x \cos(x^2) dx$

(iii) $\int \frac{x}{x^2+1} dx$

(iv) $\int x^2 e^x dx$

(v) $\int \log_e x dx$

(a) Substitution (using $u=x^2$)

(b) Integration by Parts

(c) Substitution (using $u=x^2+1$)

(d) Integration by Parts (multiple times)

(e) Integration by Parts

(i) $\int \sin^3 x \cos x dx$

(ii) $\int \frac{e^{\tan^{-1} x}}{1+x^2} dx$

(iii) $\int \frac{\log_e x}{x} dx$

(iv) $\int \frac{x^2}{(x^3+1)^4} dx$

(v) $\int \sqrt{a^2-x^2} dx$

(a) $\log_e x$

(b) $x^3+1$

(c) $\sin x$

(d) $a \sin \theta$ (Trig substitution)

(e) $\tan^{-1} x$}

(i) $\int x \sin x dx$

(ii) $\int x^2 e^x dx$

(iii) $\int \log_e x dx$

(iv) $\int x \log_e x dx$

(v) $\int e^x \cos x dx$

(a) $u=\log_e x, dv=x dx$

(b) $u=x^2, dv=e^x dx$

(c) $u=x, dv=\sin x dx$

(d) $u=\log_e x, dv=dx$

(e) $u=\cos x, dv=e^x dx$ (or vice versa, requires repeating IP)

(i) $\int [f(x)]^n f'(x) dx$

(ii) $\int \frac{f'(x)}{f(x)} dx$

(iii) $\int e^x (f(x) + f'(x)) dx$

(iv) $\int \sin(f(x)) f'(x) dx$

(v) $\int e^{f(x)} f'(x) dx$

(a) $\frac{[f(x)]^{n+1}}{n+1} + C$ ($n \neq -1$)

(b) $e^{f(x)} + C$

(c) $e^x f(x) + C$

(d) $\log_e |f(x)| + C$

(e) $-\cos(f(x)) + C$

(i) $\int \frac{x}{\sqrt{x^2+1}} dx$ (Substitution $u=x^2+1$)

(ii) $\int x \cos x dx$ (Integration by Parts)

(iii) $\int \tan x dx$ (Substitution $u=\cos x$)

(iv) $\int e^x (\sec x + \sec x \tan x) dx$ (Standard Form)

(v) $\int x e^{x^2} dx$ (Substitution $u=x^2$)

(a) $e^x \sec x + C$

(b) $\frac{1}{2} e^{x^2} + C$

(c) $\sqrt{x^2+1} + C$

(d) $x \sin x + \cos x + C$

(e) $-\log_e |\cos x| + C$

(i) $(ax+b)(cx+d)$ (distinct linear factors)

(ii) $(ax+b)^n$ (repeated linear factor)

(iii) $(x^2+ax+b)$ (irreducible quadratic factor)

(iv) $(x^2+ax+b)^n$ (repeated irreducible quadratic factor)

(v) $(ax+b)(x^2+cx+d)$ (linear and irreducible quadratic factors)

(a) $\frac{A}{ax+b} + \frac{Bx+C}{x^2+cx+d}$

(b) $\frac{A}{ax+b} + \frac{B}{cx+d}$

(c) $\frac{A}{ax+b} + \frac{B}{(ax+b)^2} + \dots + \frac{N}{(ax+b)^n}$

(d) $\frac{Ax+B}{x^2+ax+b}$

(e) $\frac{Ax+B}{x^2+ax+b} + \dots + \frac{Px+Q}{(x^2+ax+b)^n}$

(i) $\int \frac{dx}{a^2 + x^2}$

(ii) $\int \frac{dx}{a^2 - x^2}$

(iii) $\int \frac{dx}{x^2 - a^2}$

(iv) $\int \frac{dx}{\sqrt{a^2 - x^2}}$

(v) $\int \frac{dx}{\sqrt{x^2 + a^2}}$

(a) $\frac{1}{2a} \log_e |\frac{x-a}{x+a}| + C$

(b) $\log_e |x + \sqrt{x^2 + a^2}| + C$

(c) $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$

(d) $\sin^{-1}(\frac{x}{a}) + C$

(e) $\frac{1}{2a} \log_e |\frac{a+x}{a-x}| + C$

(i) $\int \frac{dx}{\sqrt{x^2 - a^2}}$

(ii) $\int \sqrt{a^2 - x^2} dx$}

(iii) $\int \sqrt{x^2 - a^2} dx$}

(iv) $\int \sqrt{x^2 + a^2} dx$}

(v) $\int \frac{dx}{x \sqrt{x^2 - a^2}}$

(a) $\frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}| + C$

(b) $\log_e |x + \sqrt{x^2 - a^2}| + C$

(c) $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$

(d) $\frac{1}{a} \sec^{-1}(\frac{x}{a}) + C$

(e) $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2} \log_e |x + \sqrt{x^2 + a^2}| + C$

(i) $\int \frac{dx}{x^2 + 4x + 5}$

(ii) $\int \frac{dx}{\sqrt{x^2 - 6x + 10}}$

(iii) $\int \sqrt{x^2 + 2x + 2} dx$

(iv) $\int \frac{dx}{\sqrt{3 - 2x - x^2}}$

(v) $\int \frac{dx}{1 + \sin x}$

(a) Complete the square to get $\sqrt{a^2 - (x+b)^2}$ form

(b) Complete the square to get $x^2+a^2$ form in denominator

(c) Use $t = \tan(x/2)$ substitution

(d) Complete the square to get $\sqrt{(x-b)^2 + a^2}$ form

(e) Complete the square to get $\sqrt{(x-b)^2 + a^2}$ form under sqrt

(i) $\int \frac{dx}{a+b\cos x}$

(ii) $\int \frac{dx}{a+b\sin x}$

(iii) $\int \frac{dx}{a\sin x + b\cos x}$

(iv) $\int \frac{dx}{\sin x}$

(v) $\int \sec x dx$

(a) Use $t = \tan(x/2)$

(b) Use $t = \tan(x/2)$

(c) Use $t = \tan(x/2)$

(d) Use $t = \tan(x/2)$ or multiply by $\frac{\sec x + \tan x}{\sec x + \tan x}$}

(e) Use $t = \tan(x/2)$ or multiply by $\frac{\text{cosec } x - \cot x}{\text{cosec } x - \cot x}$}

(i) Definite Integral $\int_a^b f(x) dx$

(ii) Lower limit of integration

(iii) Upper limit of integration

(iv) Integrand in a definite integral

(v) Geometric meaning of $\int_a^b f(x) dx$ for $f(x) \geq 0$

(a) The area under the curve from $a$ to $b$

(b) $b$

(c) $f(x)$

(d) The limit of a Riemann sum

(e) $a$}

(i) If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.

(ii) If $F$ is an antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$.

(iii) $\frac{d}{dx} \int_a^x f(t) dt$

(iv) $[F(x)]_a^b$

(v) Connects differential calculus and integral calculus.

(a) Fundamental Theorem of Calculus (Part 2 - Evaluation)

(b) Fundamental Theorem of Calculus (Part 1)

(c) Fundamental Theorem of Calculus (Part 1)

(d) Fundamental Theorem of Calculus (Part 2 - Evaluation)

(e) The entire Fundamental Theorem of Calculus

(i) $\int_0^1 x dx$

(ii) $\int_1^2 x^2 dx$

(iii) $\int_0^{\pi/2} \sin x dx$

(iv) $\int_0^1 e^x dx$

(v) $\int_1^e \frac{1}{x} dx$

(a) 1

(b) $e-1$

(c) $7/3$}

(d) 1

(e) 1/2$}

(i) Change of limits

(ii) Zero width interval

(iii) Splitting the interval

(iv) Constant multiple

(v) Sum/Difference

(a) $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$

(b) $\int_a^b k f(x) dx = k \int_a^b f(x) dx$

(c) $\int_a^a f(x) dx = 0$

(d) $\int_a^b f(x) dx = -\int_b^a f(x) dx$

(e) $\int_a^b (f(x) \pm g(x)) dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx$

(i) $\int_a^b f(x) dx$

(ii) $\int_0^4 x dx$

(iii) $\int_0^2 x^2 dx$

(iv) $\int_0^1 e^x dx$

(v) $\int_0^\pi \sin x dx$

(a) Area under $y=e^x$ from 0 to 1

(b) Area under $y=x$ from 0 to 4

(c) Area under the curve $y=f(x)$ from $x=a$ to $x=b$ and above the x-axis.

(d) Area under $y=\sin x$ from 0 to $\pi$

(e) Area under $y=x^2$ from 0 to 2

(i) $\int_0^1 (x^2+x) dx$

(ii) $\int_{-1}^1 x^3 dx$

(iii) $\int_0^1 \frac{1}{1+x^2} dx$

(iv) $\int_{-1}^1 |x| dx$

(v) $\int_0^{\pi/2} \sin^2 x dx$

(a) 1

(b) $\pi/4$

(c) 0

(d) 5/6

(e) $\pi/4$

(i) $\int_0^1 x e^{x^2} dx$

(ii) $\int_0^{\pi/2} \sin^3 x \cos x dx$

(iii) $\int_0^1 \frac{\tan^{-1} x}{1+x^2} dx$}

(iv) $\int_1^e \frac{(\log_e x)^2}{x} dx$}

(v) $\int_0^1 \sqrt{1-x^2} dx$}

(a) $x = \sin \theta$ (limits $0$ to $\pi/2$)

(b) $\tan^{-1} x$ (limits $0$ to $\pi/4$)

(c) $\sin x$ (limits $0$ to 1)

(d) $x^2$ (limits $0$ to 1)

(e) $\log_e x$ (limits $0$ to 1)

(i) $\int_{-a}^a f(x) dx = 0$

(ii) $\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$

(iii) $f(-x) = -f(x)$

(iv) $f(-x) = f(x)$

(v) Integral over $[-a, a]$ is zero

(a) $f(x)$ is an even function

(b) $f(x)$ is an odd function

(c) $f(x)$ is an odd function

(d) $f(x)$ is an even function

(e) $f(x)$ is an odd function (if the integral is defined)

(i) $\int_0^{\pi/2} \log_e(\tan x) dx$

(ii) $\int_{-\pi/2}^{\pi/2} x \cos x dx$

(iii) $\int_0^{2\pi} |\sin x| dx$

(iv) $\int_0^\pi x \sin x dx$

(v) $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$

(a) $\int_0^a f(x) dx = \int_0^a f(a-x) dx$

(b) Integral of an odd function over a symmetric interval $[-a, a]$

(c) Split the interval and use properties of modulus function

(d) Integration by parts or $\int_0^a x f(x) dx = \int_0^a (a-x) f(a-x) dx$

(e) $\int_0^a f(x) dx = \int_0^a f(a-x) dx$

(i) $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$

(ii) $\int_{-\pi/2}^{\pi/2} \sin x dx$

(iii) $\int_0^{\pi/2} \log_e(\tan x) dx$

(iv) $\int_0^\pi x \sin x dx$

(v) $\int_0^1 \frac{dx}{1+x^2}$

(a) $\pi$

(b) $\pi/4$

(c) 0

(d) 0

(e) $\pi/4$

(i) $y = x^2$, x-axis, $x=0$, $x=2$

(ii) $y = \sin x$, x-axis, $x=0$, $x=\pi$

(iii) $y = \sqrt{x}$, x-axis, $x=0$, $x=4$

(iv) $y = e^x$, x-axis, $x=0$, $x=1$

(v) $y = x$, x-axis, $x=0$, $x=5$

(a) $\int_0^4 \sqrt{x} dx$

(b) $\int_0^2 x^2 dx$

(c) $\int_0^\pi \sin x dx$

(d) $\int_0^1 e^x dx$

(e) $\int_0^5 x dx$}

(i) $y = x, y = x^2$ (between intersections)

(ii) $y = 2x, y = x$ (from $x=0$ to $x=1$)

(iii) $y = 4, y = x^2$ (between intersections)

(iv) $y = \sqrt{x}, y = x^2$ (between intersections)

(v) $y = e^x, y = e^{-x}$ (from $x=0$ to $x=1$)

(a) $\int_0^1 (e^x - e^{-x}) dx$

(b) $\int_0^1 (x - x^2) dx$

(c) $\int_0^1 (2x - x) dx$

(d) $\int_{-2}^2 (4 - x^2) dx$

(e) $\int_0^1 (\sqrt{x} - x^2) dx$

(i) Area under $y=x^2$ from 0 to 2

(ii) Area under $y=\sin x$ from 0 to $\pi$

(iii) Area between $y=x$ and $y=x^2$ from $x=0$ to $x=1$

(iv) Area between $y=x^2$ and $y=4$

(v) Area of circle $x^2+y^2=a^2$

(a) $\pi a^2$

(b) $1/6$}

(c) 2

(d) $8/3$}

(e) $32/3$}

(i) Total production from rate $r(t)$ over $[t_1, t_2]$

(ii) Change in velocity from acceleration $a(t)$ over $[t_1, t_2]$

(iii) Displacement from velocity $v(t)$ over $[t_1, t_2]$

(iv) Total amount flowed from flow rate $r(t)$ over $[t_1, t_2]$

(v) Change in total revenue from marginal revenue $R'(x)$ over $[x_1, x_2]$

(a) $\int_{t_1}^{t_2} a(t) dt$

(b) $\int_{x_1}^{x_2} R'(x) dx$

(c) $\int_{t_1}^{t_2} r(t) dt$

(d) $\int_{t_1}^{t_2} v(t) dt$

(e) $\int_{t_1}^{t_2} r(t) dt$

(i) $x = y^2$, y-axis, $y=0$, $y=1$

(ii) $x = y$, y-axis, $y=0$, $y=2$ (triangle)

(iii) $x = y^2/4$, y-axis, $y=0$, $y=2$

(iv) $x^2+y^2 = a^2$, y-axis, $y=-a$, $y=a$ (right half-circle)

(v) $x = y^2$, $x = y$ (between intersections)

(a) $\int_0^1 (y - y^2) dy$

(b) $\int_0^1 y^2 dy$

(c) $\int_0^2 \frac{y^2}{4} dy$

(d) $\int_{-a}^a \sqrt{a^2 - y^2} dy$

(e) $\int_0^2 y dy$

(i) $\frac{dy}{dx} = x^2 + y^2$

(ii) $\frac{d^2 y}{dx^2} + y = 0$

(iii) $(\frac{dy}{dx})^3 + y = x$

(iv) $\frac{d^3 y}{dx^3} + (\frac{d^2 y}{dx^2})^4 = x^5$

(v) $y' + y'' = y'''$}

(a) 2

(b) 3

(c) 1

(d) 3

(e) 1

(i) $\frac{dy}{dx} = x^2 + y^2$

(ii) $(\frac{d^2 y}{dx^2})^3 + y = 0$

(iii) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2 y}{dx^2}$

(iv) $(\frac{d^3 y}{dx^3}) + x (\frac{dy}{dx})^5 = 0$

(v) $y'' + (y')^4 = y^5$

(a) 1

(b) 3

(c) 2

(d) 1

(e) 1

(i) General Solution

(ii) Particular Solution

(iii) Singular Solution

(iv) Solution containing arbitrary constants equal to order

(v) Solution obtained by specific values of constants

(a) Cannot be obtained from the general solution

(b) Contains arbitrary constants

(c) Obtained using initial/boundary conditions

(d) General solution

(e) Particular solution

(i) $y = mx + c$ (m, c arbitrary)

(ii) $y = mx$ (m arbitrary)

(iii) $y = ax^2$ (a arbitrary)

(iv) $x^2 + y^2 = r^2$ (r arbitrary)

(v) $(x-a)^2 + y^2 = a^2$ (a arbitrary, circles through origin with center on x-axis)

(a) $y = x \frac{dy}{dx}$

(b) $\frac{d^2 y}{dx^2} = 0$

(c) $y^2 - x^2 = 2xy \frac{dy}{dx}$

(d) $2y = x \frac{dy}{dx}$}

(e) $x + y \frac{dy}{dx} = 0$

(i) $(\frac{dy}{dx})^2 + y = x$

(ii) $\frac{d^2 y}{dx^2} + x (\frac{dy}{dx})^3 = 0$

(iii) $\frac{d^3 y}{dx^3} + (\frac{d^2 y}{dx^2})^5 = y$

(iv) $\sqrt{y'} = y''$

(v) $y''' + (y'')^2 = y'$

(a) Order 3, Degree 1

(b) Order 1, Degree 2

(c) Order 2, Degree 2

(d) Order 3, Degree 1

(e) Order 2, Degree 1 (after squaring)

(i) $\frac{dy}{dx} = f(x) g(y)$

(ii) $\frac{dy}{dx} = f(\frac{y}{x})$

(iii) $\frac{dy}{dx} + P(x)y = Q(x)$

(iv) $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ ($aB \neq Ab$)

(v) $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ ($\frac{a}{A} = \frac{b}{B}$)

(a) Linear

(b) Homogeneous

(c) Reducible to Variable Separable (using $v=ax+by+c$ or similar)

(d) Reducible to Homogeneous (using $x=X+h, y=Y+k$)

(e) Variable Separable

(i) $\frac{dy}{dx} = \frac{x^2}{y}$

(ii) $\frac{dy}{dx} = e^{x+y}$

(iii) $(1+x^2) dy - (1+y^2) dx = 0$

(iv) $\frac{dy}{dx} = y \cot x$

(v) $x dy + y dx = 0$}

(a) $\frac{dy}{y} = \cot x dx$

(b) $\frac{dy}{y} = -\frac{dx}{x}$

(c) $y dy = x^2 dx$

(d) $\frac{dy}{e^y} = e^x dx$

(e) $\frac{dy}{1+y^2} = \frac{dx}{1+x^2}$}

(i) $\frac{dy}{dx} = \frac{y}{x}$

(ii) $\frac{dy}{dx} = \frac{y^2-x^2}{xy}$

(iii) $\frac{dy}{dx} = \frac{x+y}{x}$

(iv) $\frac{dy}{dx} = \frac{x+2y}{x-y}$

(v) $\frac{dy}{dx} = \frac{y}{x} + \sin(y/x)$

(a) $x \frac{dv}{dx} = \sin v$

(b) $x \frac{dv}{dx} = \frac{1-v^2}{v} - v = \frac{1-2v^2}{v}$}

(c) $x \frac{dv}{dx} = \frac{1+2v}{1-v} - v = \frac{1+2v-v+v^2}{1-v} = \frac{v^2+v+1}{1-v}$}

(d) $x \frac{dv}{dx} = v/v - v = 1-v$}

(e) $x \frac{dv}{dx} = 1$}

(i) $\frac{dy}{dx} = 2x$

(ii) $\frac{dy}{dx} = y$

(iii) $\frac{dy}{dx} = 1/x$

(iv) $\frac{dy}{dx} = \cos x$

(v) $\frac{dy}{dx} = e^x$

(a) $y = \sin x + C$

(b) $y = \log_e |x| + C$

(c) $y = e^x + C$

(d) $y = x^2 + C$

(e) $y = C e^x$}

(i) $\frac{dy}{dx} = (x+y)^2$

(ii) $\frac{dy}{dx} = \frac{x+y+1}{x+y-1}$

(iii) $\frac{dy}{dx} = \sin(x-y)$

(iv) $\frac{dy}{dx} = \frac{2x+4y+1}{x+2y-1}$

(v) $\frac{dy}{dx} = \tan^2(x+y-5)$

(a) $x+y$

(b) $x-y$

(c) $x+y+1$ or $x+y-1$

(d) $x-y+2$ (Doesn't fit form)

(e) $x+2y+1$ or $x+2y-1$}

(i) $y' + xy = x^2$

(ii) $x y' + y = \sin x$

(iii) $(1+x^2) \frac{dy}{dx} + 2xy = 1/x$

(iv) $y dx - x dy = 0$ (Not linear in y)

(v) $\frac{dx}{dy} - x/y = y^2$}

(a) $\frac{dy}{dx} + \frac{2x}{1+x^2} y = \frac{1}{x(1+x^2)}$

(b) $\frac{dx}{dy} + (-1/y) x = y^2$}

(c) $\frac{dy}{dx} + x y = x^2$

(d) $\frac{dy}{dx} + (1/x) y = \sin x/x$}

(e) $\frac{dy}{dx} + (-y/x) = 0$

(i) $y' + y = x$

(ii) $y' + (1/x) y = x^2$

(iii) $y' + y \tan x = \sec x$

(iv) $y' - y = e^x$

(v) $y' + (2x/(x^2+1)) y = 1$}

(a) $e^x$

(b) $x$

(c) $\sec x$}

(d) $e^{-x}$

(e) $x^2+1$}

(i) $y' + y = x$ (IF = $e^x$)

(ii) $y' + (1/x) y = x^2$ (IF = $x$)

(iii) $y' + y \tan x = \sec x$ (IF = $\sec x$)

(iv) $y' - y = e^x$ (IF = $e^{-x}$)

(v) $y' + (2x/(x^2+1)) y = 1$ (IF = $x^2+1$)

(a) $(y x)' = x^3$

(b) $(y e^x)' = x e^x$

(c) $(y \sec x)' = \sec^2 x$

(d) $(y e^{-x})' = 1$

(e) $(y (x^2+1))' = x^2+1$}

(i) $\frac{dx}{dy} + x = y$

(ii) $\frac{dx}{dy} - x = e^y$

(iii) $\frac{dx}{dy} + (1/y) x = y^2$

(iv) $\frac{dx}{dy} - 2y x = y^3$

(v) $\frac{dx}{dy} + x \cot y = \sin y$

(a) $e^{\int \cot y dy} = \sin y$}

(b) $e^{\int -1 dy} = e^{-y}$

(c) $e^{\int 1 dy} = e^y$

(d) $e^{\int (1/y) dy} = y$}

(e) $e^{\int -2y dy} = e^{-y^2}$}

(i) $y' + y = 0$

(ii) $y' = x$

(iii) $y' = y+1$

(iv) $y' + y/x = 0$

(v) $y' = e^x$

(a) $y = C e^x$}

(b) $y = x^2/2 + C$

(c) $y = C/x$}

(d) $y = -1 + C e^x$}

(e) $y = e^x + C$

(i) Simple Population Growth (unlimited resources)

(ii) Radioactive Decay

(iii) Compound Interest (continuous)

(iv) Newton's Law of Cooling (Temp difference $T-T_a$)

(v) Velocity of falling object with air resistance $\propto v$

(a) $\frac{dA}{dt} = rA$

(b) $\frac{dP}{dt} = kP$

(c) $\frac{dN}{dt} = -kN$

(d) $\frac{dT}{dt} = -k(T-T_a)$

(e) $m \frac{dv}{dt} = mg - kv$ (Linear DE in v)

(i) Growth rate constant

(ii) Decay constant

(iii) Ambient temperature

(iv) Carrying capacity (in logistic growth)

(v) Interest rate (continuous compound interest)

(a) $k$ in $\frac{dP}{dt} = kP$

(b) $k$ in $\frac{dN}{dt} = -kN$

(c) $T_a$ in $\frac{dT}{dt} = -k(T-T_a)$

(d) $M$ in $\frac{dP}{dt} = kP(M-P)$

(e) $r$ in $\frac{dA}{dt} = rA$

(i) $\frac{dy}{dx} = 2x, y(0)=3$

(ii) $\frac{dy}{dt} = -y, y(0)=5$

(iii) $\frac{dA}{dt} = 0.05A, A(0)=10000$

(iv) $\frac{dT}{dt} = -(T-20), T(0)=100$

(v) $\frac{dv}{dt} = 9.8, v(0)=0$

(a) $y(x) = x^2+3$

(b) $y(t) = 5e^{-t}$

(c) $A(t) = 10000 e^{0.05t}$

(d) $T(t) = 20 + 80e^{-t}$

(e) $v(t) = 9.8t$

(i) Marginal Cost

(ii) Total Cost

(iii) Marginal Revenue

(iv) Total Revenue

(v) Profit

(a) Can be found by integrating Marginal Revenue

(b) Can be found by integrating Marginal Cost

(c) Derivative of Total Cost

(d) Derivative of Total Revenue

(e) Difference between Total Revenue and Total Cost

(i) Rate of change proportional to the quantity.

(ii) Rate of change proportional to the quantity, plus a constant source/sink.

(iii) Rate of change proportional to the product of the quantity and its difference from a carrying capacity.

(iv) Rate of change of temperature difference proportional to the difference.

(v) Velocity change due to gravity and resistance proportional to velocity.

(a) $\frac{dy}{dt} = ky(M-y)$

(b) $\frac{dy}{dt} = ky$

(c) $\frac{dy}{dt} = ky + c$ (Linear DE)

(d) $\frac{dy}{dt} = -k(y-y_a)$

(e) $\frac{dv}{dt} = g - kv/m$ (Linear DE)