AP Calculus AB Exam Study Guide Notes

Table of Contents

• INTRODUCTION
• TOPICS TO STUDY
  • Elementary Functions
  • Limits
  • Differential Calculus
  • Integral Calculus
• SOME USEFUL FORMULAS
• CALCULATOR TIPS AND PROGRAMS
• BOOK REVIEW OF AVAILABLE STUDY GUIDES
• ACKNOWLEDGEMENTS

Introduction

• Advanced Placement (AP) is a program offering college-level courses and exams.
• The AP Calculus AB Exam assesses introductory differential and integral calculus skills.
• It covers a full-year college mathematics course.
• Exam Sections:
  • Multiple Choice Part A: 25 questions, 45 minutes, no calculator.
  • Multiple Choice Part B: 15 questions, 45 minutes, graphing calculator required.
  • Free Response: 6 questions, 45 minutes, graphing calculator required.
• Scoring:
  • Multiple choice and free response sections have equal weight.
  • Grades are reported on a scale of 1 to 5.
    • 5: Extremely well qualified
    • 4: Well qualified
    • 3: Qualified
    • 2: Possibly qualified
    • 1: No recommendation
  • A score of 3 or higher requires approximately 50% correct answers on multiple-choice questions along with acceptable free-response work.
  • Multiple-choice penalty: 1/4 point is deducted for each incorrect answer.

Topics to Study

Elementary Functions

• Properties of Functions
  • A function $ƒ$ is a set of ordered pairs $(x, y)$, where each element $x$ corresponds to exactly one element $y$.
    • Domain: the set of all $x$ values.
    • Range: the set of all $y$ values.
• Combinations of Functions
  • If $ƒ(x) = 3x + 1$ and $g(x) = x^2 - 1$, then:
    • Sum: $ƒ(x) + g(x) = (3x + 1) + (x^2 - 1) = x^2 + 3x$
    • Difference: $ƒ(x) - g(x) = (3x + 1) - (x^2 - 1) = -x^2 + 3x + 2$
    • Product: $ƒ(x)g(x) = (3x + 1)(x^2 - 1) = 3x^3 + x^2 - 3x - 1$
    • Quotient: $ƒ(x)/g(x) = (3x + 1)/(x^2 - 1)$
    • Composite: $(ƒ \,°\, g)(x) = ƒ(g(x)) = 3(x^2 - 1) + 1 = 3x^2 - 2$
• Inverse Functions
  • Functions $ƒ$ and $g$ are inverses if $ƒ(g(x)) = x$ for all $x$ in the domain of $g$ and $g(ƒ(x)) = x$ for all $x$ in the domain of $ƒ$.
  • The inverse of $ƒ$ is denoted $ƒ^{-1}$.
  • To find $ƒ^{-1}$, switch $x$ and $y$ in the original equation and solve for $y$.
  • Example: If $ƒ(x) = 3x + 2$, then $ƒ^{-1}(x) = (x-2)/3$
• Even and Odd Functions
  • Even: $y = ƒ(x)$ is even if $ƒ(-x) = ƒ(x)$; symmetric about the y-axis (e.g., $y = x^2$).
  • Odd: $y = ƒ(x)$ is odd if $ƒ(-x) = -ƒ(x)$; symmetric about the origin (e.g., $y = x^3$).
• Periodic Functions
  • Familiarity with definitions and graphs of trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) is expected.
  • Example: If $ƒ(x) = \sin(\tan^{-1} x)$, the range of $ƒ$ is $(-1, 1)$.
• Zeros of a Function
  • Occur where the function $ƒ(x)$ crosses the x-axis (also called roots).
  • Example: The zeros of $ƒ(x) = x^3 - 2x^2 + x$ are $0$ and $1$. Factoring gives $ƒ(x) = x(x^2 - 2x + 1) = x(x - 1)^2$
• Properties of Graphs
  • Intercepts
  • Symmetry
  • Asymptotes
  • Transformations:
    • $y = kƒ(x)$
    • $y = ƒ(kx)$
    • $y - k = ƒ(x - h)$
    • $y = |ƒ(x)|$
    • $y = ƒ(|x|)$

Limits

• Properties of Limits
  • If $b$ and $c$ are real numbers, $n$ is a positive integer, and $ƒ$ and $g$ have limits as $x → c$, then:
    1. Scalar Multiple: $\lim<em>{x→c} [b(ƒ(x))] = b[\lim</em>{x→c} ƒ(x)]$
    2. Sum or Difference: $\lim<em>{x→c} [ƒ(x) ± g(x)] = \lim</em>{x→c} ƒ(x) ± \lim_{x→c} g(x)$
    3. Product: $\lim<em>{x→c} [ƒ(x)g(x)] = [\lim</em>{x→c} ƒ(x)][\lim_{x→c} g(x)]$
    4. Quotient: $\lim<em>{x→c} [ƒ(x)/g(x)] = [\lim</em>{x→c} ƒ(x)]/[\lim<em>{x→c} g(x)]$ if $\lim</em>{x→c} g(x) ≠ 0$
• One-Sided Limits
  • $\lim_{x→a^+} ƒ(x)$ approaches $a$ from the right.
  • $\lim_{x→a^-} ƒ(x)$ approaches $a$ from the left.
• Limits at Infinity
  • $\lim<em>{x→+\infty} ƒ(x) = L$ or $\lim</em>{x→-\infty} ƒ(x) = L$: The value of $ƒ(x)$ approaches $L$ as $x$ increases/decreases without bound.
  • $y = L$ is the horizontal asymptote of the graph of $ƒ$.
• Some Nonexistent Limits
  • $\lim_{x→0} \frac{1}{x^2}$
  • $\lim_{x→0} \frac{|x|}{x}$
  • $\lim_{x→0} \sin(\frac{1}{x})$
• Some Infinite Limits
  • $\lim_{x→0} \frac{1}{x^2} = \infty$
  • $\lim_{x→0^+} \ln x = -\infty$
• Limit Example
  • $\lim_{x→0} \frac{\sin x}{x} = 1$ (memorize this limit)
• Continuity
  • Definition: A function $ƒ$ is continuous at $c$ if:
    1. $ƒ(c)$ is defined
    2. $\lim_{x→c} ƒ(x)$ exists
    3. $\lim_{x→c} ƒ(x) = ƒ(c)$
  • Graphically, a function is continuous at $c$ if the graph can be traced through $(c, ƒ(c))$ without lifting the pencil.
  • Example:
    • If $f(x) = \begin{cases} \frac{\sqrt{3+x} - 2}{x} \text{, for } x \neq 0 \ k \text{, for } x = 0 \end{cases}$
    • and if $ƒ$ is continuous at $x = 0$, then $k = 3/2$.
    • $\lim_{x→0} ƒ(x) = 3/2$
• Intermediate Value Theorem
  • If $ƒ$ is continuous on $[a, b]$ and $k$ is any number between $ƒ(a)$ and $ƒ(b)$, then there is at least one number $c$ between $a$ and $b$ such that $ƒ(c) = k$.

Differential Calculus

• Definition
  • $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ if this limit exists
  • $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$
  • If $ƒ$ is differentiable at $x = c$, then $ƒ$ is continuous at $x = c$.
• Differentiation Rules
  • General and Logarithmic Differentiation Rules
    1. $\frac{d}{dx} [cu] = cu'$
    2. $\frac{d}{dx} [u ± v] = u' ± v'$ (sum rule)
    3. $\frac{d}{dx} [uv] = uv' + vu'$ (product rule)
    4. $\frac{d}{dx} [\frac{u}{v}] = \frac{vu' - uv'}{v^2}$ (quotient rule)
    5. $\frac{d}{dx} [c] = 0$
    6. $\frac{d}{dx} [u^n] = nu^{n-1}u'$ (power rule)
    7. $\frac{d}{dx} [x] = 1$
    8. $\frac{d}{dx} [\ln u] = \frac{u'}{u}$
    9. $\frac{d}{dx} [e^u] = e^uu'$
    10. $\frac{d}{dx} [ƒ(g(x))] = ƒ'(g(x)) g'(x)$ (chain rule)
• Derivatives of Trigonometric Functions
  1. $\frac{d}{dx} [\sin u] = (\cos u)u'$
  2. $\frac{d}{dx} [\csc u] = -(\csc u \cot u)u'$
  3. $\frac{d}{dx} [\cos u] = -(\sin u)u'$
  4. $\frac{d}{dx} [\sec u] = (\sec u \tan u)u'$
  5. $\frac{d}{dx} [\tan u] = (\sec^2 u)u'$
  6. $\frac{d}{dx} [\cot u] = -(\csc^2 u)u'$
• Derivatives of Inverse Trigonometric Functions
  1. $\frac{d}{dx} [\arcsin u] = \frac{u'}{\sqrt{1 - u^2}}$
  2. $\frac{d}{dx} [\operatorname{arccsc} u] = \frac{-u'}{|u|\sqrt{u^2 - 1}}$
  3. $\frac{d}{dx} [\arccos u] = \frac{-u'}{\sqrt{1 - u^2}}$
  4. $\frac{d}{dx} [\operatorname{arcsec} u] = \frac{u'}{|u|\sqrt{u^2 - 1}}$
  5. $\frac{d}{dx} [\arctan u] = \frac{u'}{1 + u^2}$
  6. $\frac{d}{dx} [\operatorname{arccot} u] = \frac{-u'}{1 + u^2}$
• Implicit Differentiation
  • Useful when you cannot easily solve for $y$ as a function of $x$.
• Higher Order Derivatives
  • Successive derivatives of $ƒ(x)$.
  • $ƒ''(x)$ is the second derivative (derivative of $ƒ'(x)$).
  • Numerical notation: $ƒ^{(n)}(x) = y^{(n)}$
  • Also indicated by $\frac{d^2y}{dx^2}$.
  • Example: If $y = x^5$, then $y' = 5x^4$, $y'' = 20x^3$, $y''' = 60x^2$
• Derivatives of Inverse Functions
  • If $y = ƒ(x)$ and $x = ƒ^{-1}(y)$ are differentiable inverse functions, then their derivatives are reciprocals: $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$
• Logarithmic Differentiation
  1. Take $\ln$ of both sides
  2. Differentiate
  3. Solve for $y'$
  4. Substitute for $y$
  5. Simplify
• Mean Value Theorem
  • If $ƒ$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a number $c$ in $(a, b)$ such that $ƒ'(c) = \frac{f(b) - f(a)}{b - a}$.
• L'Hôpital's Rule
  • If $\lim \frac{ƒ(x)}{g(x)}$ is an indeterminate form of $0/0$ or $\infty / \infty$, and if $\lim \frac{ƒ'(x)}{g'(x)}$ exists, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$.
  • The indeterminate form $0 \cdot \infty$ can be reduced to $0/0$ or $\infty / \infty$ to apply L'Hôpital's Rule.
  • L'Hôpital's Rule can be applied to the four different indeterminate forms of $\infty / \infty$: $\frac{\infty}{\infty}$, $\frac{-\infty}{\infty}$, $\frac{\infty}{-\infty}$, and $\frac{-\infty}{-\infty}$.
  • Example: $\lim_{x \to 0} \frac{\sin x}{x + 1} = 1$
• Tangent and Normal Lines
  • The derivative of a function at a point is the slope of the tangent line.
  • The normal line is perpendicular to the tangent line at the point of tangency.
  • Example: The slope of the normal line to the curve $y = 2x^2 + 1$ at $(1, 3)$ is $-1/4$.
• Extreme Value Theorem
  • If a function $ƒ(x)$ is continuous on a closed interval, then $ƒ(x)$ has both a maximum and minimum value in the interval.
• Curve Sketching
  • $f'(c) > 0$: $ƒ$ increasing at $c$
  • $f'(c) < 0$: $ƒ$ decreasing at $c$
  • $f'(c) = 0$: Horizontal tangent at $c$
  • $f'(c) = 0, f'(c^-) < 0, f'(c^+) > 0$: Relative minimum at $c$
  • $f'(c) = 0, f'(c^-) > 0, f'(c^+) < 0$: Relative maximum at $c$
  • $f'(c) = 0, f''(c) > 0$: Relative minimum at $c$
  • $f'(c) = 0, f''(c) < 0$: Relative maximum at $c$
  • $f'(c) = 0, f''(c) = 0$: Further investigation required
  • $f''(c) > 0$: Concave upward
  • $f''(c) < 0$: Concave downward
  • $f''(c) = 0$: Further investigation required
  • $f''(c) = 0, f''(c^-) < 0, f''(c^+) > 0$: Point of inflection
  • $f''(c) = 0, f''(c^-) > 0, f''(c^+) < 0$: Point of inflection
  • $f(c)$ exists, $f'(c)$ does not exist: Possibly a vertical tangent; possibly an absolute max. or min.
• Newton's Method for Approximating Zeros of a Function
  • $x<em>{n + 1} = x</em>n - \frac{f(x<em>n)}{f'(x</em>n)}$
  • Let $x_1$ be a guess for one of the roots. Reiterate the function with the result until the required accuracy is obtained.
• Optimization Problems
  • Calculus can be used to solve practical problems requiring maximum or minimum values.
• Rates-of-Change Problems
  • Distance, Velocity, and Acceleration
    • $y = s(t)$: Position of a particle along a line at time $t$
    • $v = s'(t)$: Instantaneous velocity (rate of change) at time $t$
    • $a = v'(t) = s''(t)$: Instantaneous acceleration at time $t$
• Related Rates of Change
  • Calculus can be used to find the rate of change of two or more variables that are functions of time $t$ by differentiating with respect to $t$.

Integral Calculus

• Indefinite Integrals
  • Definition: A function $F(x)$ is the antiderivative of a function $ƒ(x)$ if for all $x$ in the domain of $ƒ$, $F'(x) = ƒ(x)$
  • $\int ƒ(x) dx = F(x) + C$, where $C$ is a constant.
• Basic Integration Formulas
  • General and Logarithmic Integrals
    1. $\int kƒ(x) dx = k \int ƒ(x) dx$
    2. $\int [ƒ(x) ± g(x)] dx = \int ƒ(x) dx ± \int g(x) dx$
    3. $\int k dx = kx + C$
    4. $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n ≠ -1$
    5. $\int e^x dx = e^x + C$
    6. $\int a^x dx = \frac{a^x}{\ln a} + C$, $a > 0, a ≠ 1$
    7. $\int \frac{dx}{x} = \ln |x| + C$
  • Trigonometric Integrals
    1. $\int \sin x dx = -\cos x + C$
    2. $\int \cos x dx = \sin x + C$
    3. $\int \sec^2 x dx = \tan x + C$
    4. $\int \csc^2 x dx = -\cot x + C$
    5. $\int \sec x \tan x dx = \sec x + C$
    6. $\int \csc x \cot x dx = -\csc x + C$
    7. $\int \tan x dx = -\ln |\cos x| + C$
    8. $\int \cot x dx = \ln |\sin x| + C$
    9. $\int \sec x dx = \ln |\sec x + \tan x| + C$
    10. $\int \csc x dx = -\ln |\csc x + \cot x| + C$
    11. $\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C$
    12. $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan \frac{x}{a} + C$
    13. $\int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \operatorname{arcsec} \frac{x}{a} + C$
• Integration by Substitution
  • $\int ƒ(g(x))g'(x) dx = F(g(x)) + C$
  • If $u = g(x)$, then $du = g'(x) dx$ and $\int ƒ(u) du = F(u) + C$
• Integration by Parts
  • $\int u dv = uv - \int v du$
• Distance, Velocity, and Acceleration (on Earth)
  • $a(t) = s''(t) = -32 \text{ ft/sec}^2$
  • $v(t) = s'(t) = \int s''(t) dt = \int -32 dt = -32t + C_1$
    • at $t = 0$, $v<em>0 = v(0) = (-32)(0) + C</em>1 = C_1$
  • $s(t) = \int v(t) dt = \int (-32t + v<em>0) dt = -16t^2 + v</em>0t + C_2$
• Separable Differential Equations
  • It is sometimes possible to separate variables and write a differential equation in the form $ƒ(y) dy + g(x) dx = 0$ by integrating: $\int ƒ(y) dy + \int g(x) dx = C$
• Applications to Growth and Decay
  • Often, the rate of change or a variable $y$ is proportional to the variable itself.
    • $\frac{dy}{dt} = ky$
    • Separate the variables: $\frac{dy}{y} = k dt$
    • Integrate both sides: $\ln |y| = kt + C_1$
    • $y = Ce^{kt}$ (Law of Exponential Growth and Decay)
      • Exponential growth when $k > 0$
      • Exponential decay when $k < 0$
• Definition of the Definite Integral
  • The definite integral is the limit of the Riemann sum of $ƒ$ on the interval $[a, b]$
    • $\lim<em>{\Delta x \to 0} \sum</em>{i=1}^n f(x<em>i) \Delta x = \int</em>a^b ƒ(x) dx$
• Properties of Definite Integrals
  1. $\int<em>a^b [ƒ(x) + g(x)] dx = \int</em>a^b ƒ(x) dx + \int_a^b g(x) dx$
  2. $\int<em>a^b kƒ(x) dx + k \int</em>a^b ƒ(x) dx$
  3. $\int_a^a ƒ(x) dx = 0$
  4. $\int<em>a^b ƒ(x) dx = -\int</em>b^a ƒ(x) dx$
  5. $\int<em>a^b ƒ(x) dx + \int</em>b^c ƒ(x) dx = \int_a^c ƒ(x) dx$
  6. If $ƒ(x) ≤ g(x)$ on $[a, b]$, then $\int<em>a^b ƒ(x) dx ≤ \int</em>a^b g(x) dx$
• Approximations to the Definite Integral
  • Riemann Sums
    • $\int<em>a^b ƒ(x)dx = S</em>n = \sum<em>{i=1}^n f(x</em>i) \Delta x$
  • Trapezoidal Rule
    • $\int<em>a^b ƒ(x)dx ≈ [\frac{1}{2}ƒ(x</em>0) + ƒ(x<em>1) + ƒ(x</em>2) + … + ƒ(x<em>{n-1}) + \frac{1}{2}ƒ(x</em>n)] \frac{b - a}{n}$
• The Fundamental Theorem of Calculus
  • If $ƒ$ is continuous on $[a, b]$ and if $F' = ƒ$, then $\int_a^b ƒ(x) dx = F(b) - F(a)$
• The Second Fundamental Theorem of Calculus
  • If $ƒ$ is continuous on an open interval $I$ containing $a$, then for every $x$ in the interval, $\frac{d}{dx} \int_a^x ƒ(t) dt = ƒ(x)$
• Area Under a Curve
  • If $ƒ(x) ≥ 0$ on $[a, b]$ then $A = \int_a^b ƒ(x) dx$
  • If $ƒ(x) ≤ 0$ on $[a, b]$ then $A = -\int_a^b ƒ(x) dx$
  • If $ƒ(x) ≥ 0$ on $[a, c]$ and $ƒ(x) ≤ 0$ on $[c, b]$ then $A = \int<em>a^c ƒ(x) dx - \int</em>c^b ƒ(x) dx$
• Average Value of a Function on an Interval
  • $\frac{1}{b - a} \int_a^b ƒ(x) dx$
• Volumes of Solids with Known Cross Sections
  1. For cross-sections of area $A(x)$, taken perpendicular to the x-axis: $V = \int_a^b A(x) dx$
  2. For cross-sections of area $A(y)$, taken perpendicular to the y-axis: $V = \int_a^b A(y) dy$
• Volumes of Solids of Revolution: Disk Method
  • $V = \int_a^b πr^2 dx$
  • Rotated about the x-axis: $V = \int_a^b π[ƒ(x)]^2 dx$
  • Rotated about the y-axis: $V = \int_a^b π[ƒ(y)]^2 dy$
• Volumes of Solids of Revolution: Washer Method
  • $V = \int<em>a^b π(r</em>o^2 dx - r_i^2) dx$
  • Rotated about the x-axis: $V = \int<em>a^b π[(ƒ</em>1(x))^2 - (ƒ_2(x))^2] dx$
  • Rotated about the y-axis: $V = \int<em>a^b π[(ƒ</em>1(y))^2 - (ƒ_2(y))^2] dy$
• Volumes of Solids of Revolution: Cylindrical Shell Method
  • $V = \int_a^b 2πrh dr$
  • Rotated about the x-axis: $V = 2π \int_a^b xƒ(x) dx$
  • Rotated about the y-axis: $V = 2π \int_a^b yƒ(y) dy$

Some Useful Formulas

• $\log_a x = \frac{\log x}{\log a}$
• $\sin^2x + \cos^2x = 1$
• $1 + \tan^2x = \sec^2x$
• $1 + \cot^2x = \csc^2x$
• $\sin 2x = 2 \sin x \cos x$
• $\cos 2x = \cos^2x - \sin^2x$
• $\sin^2x = \frac{1}{2}(1- \cos 2x)$
• $\cos^2x = \frac{1}{2}(1+ \cos 2x)$
• Volume of a right circular cylinder = $πr^2h$
• Volume of a cone = $\frac{1}{3} πr^2h$
• Volume of a sphere = $\frac{4}{3} πr^3$

Calculator Tips and Programs

• Calculator Tips
  • The TI-82 is referenced, however, most tips will apply to any graphing calculator.
  • Calculator is required for Part B of Section I and for Section II.
  • Skills to master include:
    • Simple calculations
    • Finding intersection of two graphs
    • Graphing a function and analyzing its properties (domain, range, asymptotes)
  • Key Functions in CALC Menu:
    1. Calculate value of a function at $x = c$
    2. Calculate roots of a function
    3. Find minimum of a function
    4. Find maximum of a function
    5. Find point of intersection of two functions
    6. Find slope of tangent at $(x, y)$
    7. Find area under the curve from $a$ to $b$
  • Key Functions in MATH Menu:
    6. Find the minimum of a function: fMin(expression, variable, lower, upper)
    7. Find the maximum of a function: fMax(expression, variable, lower, upper)
    8. Find the numerical derivative at a given value: nDeriv(expression, variable, value)
    9. Find the numerical integral of an expression: fnInt(expression, variable, lower, upper)
    10. Calculate the root of an expression: solve(expression, variable, guess, {lower, upper})
• Calculator Programs
  • Program to solve for f(x):
    • PROGRAM: SOLVE
    • : Input X
    • : 3x2 + 2 → X [type your function here and place → X at the end]
    • : Disp X
  • Program to solve for a quadratic equation:
    • PROGRAM: QUADRAT
    • :Input "A? ", A
    • :Input "B? ", B
    • :Input "C? ", C
    • :(- B + √ (B2 - 4AC) ) / 2A → D
    • :(- B - √ (B2 - 4AC) ) / 2A → E
    • :B2 - 4AC → F
    • :ClrHome
    • :Disp "+ EQUALS"
    • :Disp D
    • :Disp "- EQUALS"
    • :Disp E
    • :Disp "B2 - 4AC EQUALS"
    • :Disp F
    • To Run: Enter a, b, and c for ax2 + bx + c. "+ EQUALS" and "- EQUALS" give the roots of the equation
  • Program that will use the trapezoidal rule to approximate a definite integral:
    • PROGRAM: TRAP
    • :ClrHome
    • :Input "F(X) IN QUOTES:", Y0
    • :Input "START(A):", A
    • :Input "END(B):", B
    • :Input "NO. OF DIV. (N):", N
    • :(B - A) / N → D
    • :0 → S
    • :For (X, A, B, D)
    • :S + Y0 → S
    • :End
    • :A → X
    • : Y0 → F
    • :B → X
    • : Y0 → L
    • :D * (-F + S - L) → A
    • :ClrHome
    • :Disp "EST AREA="
    • :Disp A