apcalc

Introduction

  • Advanced Placement (AP) is a program offering college-level courses and exams, providing high school students the chance to earn advanced placement or college credit.
  • The AP Calculus AB Exam assesses introductory differential and integral calculus skills, equivalent to a full-year college course.
  • The exam has three sections:
    • Multiple Choice Part A: 25 questions, 45 minutes, no calculators allowed.
    • Multiple Choice Part B: 15 questions, 45 minutes, calculators required.
    • Free Response: 6 questions, 45 minutes, calculators required.
  • Both multiple-choice and free-response sections have equal weighting.
  • 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 in the multiple-choice section and acceptable work in the free-response section.
  • In the multiple-choice sections, 1/4 of incorrect answers are subtracted from the number of correct answers to account for guessing.

Topics to Study

Elementary Functions

  • Properties of Functions
    • A function ff is defined as a set of ordered pairs (x,y)(x, y), where each xx corresponds to exactly one yy.
    • The domain of ff is the set of all xx values.
    • The range of ff is the set of all yy values.
  • Combinations of Functions
    • Given f(x)=3x+1f(x) = 3x + 1 and g(x)=x2−1g(x) = x^2 - 1
      • Sum: f(x)+g(x)=(3x+1)+(x2−1)=x2+3xf(x) + g(x) = (3x + 1) + (x^2 - 1) = x^2 + 3x
      • Difference: f(x)−g(x)=(3x+1)−(x2−1)=−x2+3x+2f(x) - g(x) = (3x + 1) - (x^2 - 1) = -x^2 + 3x + 2
      • Product: f(x)g(x)=(3x+1)(x2−1)=3x3+x2−3x−1f(x)g(x) = (3x + 1)(x^2 - 1) = 3x^3 + x^2 - 3x - 1
      • Quotient: f(x)/g(x)=(3x+1)/(x2−1)f(x)/g(x) = (3x + 1)/(x^2 - 1)
      • Composite: (f∘g)(x)=f(g(x))=3(x2−1)+1=3x2−2(f \circ g)(x) = f(g(x)) = 3(x^2 - 1) + 1 = 3x^2 - 2
  • Inverse Functions
    • Functions ff and gg are inverses if f(g(x))=xf(g(x)) = x for each xx in the domain of gg and g(f(x))=xg(f(x)) = x for each xx in the domain of ff.
    • The inverse of ff is denoted f−1f^{-1}.
    • To find f−1f^{-1}, switch xx and yy in the original equation and solve for yy.
    • Exercise: If f(x)=3x+2f(x) = 3x + 2, then f−1(x)=x−23f^{-1}(x) = \frac{x-2}{3}
  • Even and Odd Functions
    • A function y=f(x)y = f(x) is even if f(−x)=f(x)f(-x) = f(x). Even functions are symmetric about the y-axis (e.g., y=x2y = x^2).
    • A function y=f(x)y = f(x) is odd if f(−x)=−f(x)f(-x) = -f(x). Odd functions are symmetric about the origin (e.g., y=x3y = x^3).
  • Reflection of Functions
    • The reflection of y=f(x)y = f(x) in the y-axis is y=f(−x)y = f(-x).
    • Exercise: The reflection of y=3x+1y = 3x + 1 about the y-axis is y=3−x+1y = 3^{-x} + 1.
  • Periodic Functions
    • Familiarity with definitions and graphs of trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) is expected.
    • Exercise: If f(x)=sin⁡(tan⁡−1x)f(x) = \sin(\tan^{-1} x), the range of ff is (−1,1)(-1, 1).
  • Zeros of a Function
    • Zeros occur where the function f(x)f(x) crosses the x-axis (also called roots).
    • Exercise: The zeros of f(x)=x3−2x2+xf(x) = x^3 - 2x^2 + x are 0 and 1, since f(x)=x(x2−2x+1)=x(x−1)2f(x) = x(x^2 - 2x + 1) = x(x - 1)^2.

Properties of Graphs

  • Intercepts
  • Symmetry
  • Asymptotes
  • Relationships between the graph of y=f(x)y = f(x) and
    • y=kf(x)y = kf(x)
    • y=f(kx)y = f(kx)
    • y−k=f(x−h)y - k = f(x - h)
    • y=∣f(x)∣y = |f(x)|
    • y=f(∣x∣)y = f(|x|)

Limits

  • Properties of Limits
    • If bb and cc are real numbers, nn is a positive integer, and the functions ff and gg have limits as x→cx \to c, then:
      1. Scalar multiple: lim⁡<em>x→c[b(f(x))]=b[lim⁡</em>x→cf(x)]\lim<em>{x \to c} [b(f(x))] = b[\lim</em>{x \to c} f(x)]
      2. Sum or difference: lim⁡<em>x→c[f(x)±g(x)]=lim⁡</em>x→cf(x)±lim⁡x→cg(x)\lim<em>{x \to c} [f(x) \pm g(x)] = \lim</em>{x \to c} f(x) \pm \lim_{x \to c} g(x)
      3. Product: lim⁡<em>x→c[f(x)g(x)]=[lim⁡</em>x→cf(x)][lim⁡x→cg(x)]\lim<em>{x \to c} [f(x)g(x)] = [\lim</em>{x \to c} f(x)][\lim_{x \to c} g(x)]
      4. Quotient: lim⁡<em>x→c[f(x)/g(x)]=[lim⁡</em>x→cf(x)]/[lim⁡<em>x→cg(x)]\lim<em>{x \to c} [f(x)/g(x)] = [\lim</em>{x \to c} f(x)]/[\lim<em>{x \to c} g(x)] if lim⁡</em>x→cg(x)≠0\lim</em>{x \to c} g(x) \neq 0
  • One-Sided Limits
    • lim⁡x→a+f(x)\lim_{x \to a^+} f(x): xx approaches cc from the right
    • lim⁡x→a−f(x)\lim_{x \to a^-} f(x): xx approaches cc from the left
  • Limits at Infinity
    • lim⁡<em>x→+∞f(x)=L\lim<em>{x \to +\infty} f(x) = L or lim⁡</em>x→−∞f(x)=L\lim</em>{x \to -\infty} f(x) = L
    • The value of f(x)f(x) approaches LL as xx increases/decreases without bound.
    • y=Ly = L is the horizontal asymptote of the graph of ff.
  • Some Nonexistent Limits
    • lim⁡x→01x2\lim_{x \to 0} \frac{1}{x^2}
    • lim⁡x→0∣x∣x\lim_{x \to 0} \frac{|x|}{x}
    • lim⁡x→0sin⁡1x\lim_{x \to 0} \sin \frac{1}{x}
  • Some Infinite Limits
    • lim⁡x→01x2=∞\lim_{x \to 0} \frac{1}{x^2} = \infty
    • lim⁡x→0+ln⁡x=−∞\lim_{x \to 0^+} \ln x = -\infty
  • Exercise: What is lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}?
    • Answer: 1 (memorize this limit)
  • Continuity Definition
    • A function ff is continuous at cc if:
      1. f(c)f(c) is defined
      2. lim⁡x→cf(x)\lim_{x \to c} f(x) exists
      3. lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)
    • Graphically, the function is continuous at cc if a pencil can be moved along the graph of f(x)f(x) through (c,f(c))(c, f(c)) without lifting it off the graph.
    • Exercise: If
      f(x)={x3+3xxamp;for x≠0 kamp;for x=0f(x) = \begin{cases} \frac{x^3 + 3x}{x} &amp; \text{for } x \neq 0 \ k &amp; \text{for } x = 0 \end{cases}
      and if ff is continuous at x=0x = 0, then k=3/2k = 3/2, because lim⁡x→0f(x)=3/2\lim_{x \to 0} f(x) = 3/2
  • Intermediate Value Theorem
    • If ff is continuous on [a,b][a, b] and kk is any number between f(a)f(a) and f(b)f(b), then there is at least one number cc between aa and bb such that f(c)=kf(c) = k.

Differential Calculus

  • Definition

    • f′(x)=lim⁡Δx→0f(x+Δx)−f(x)Δxf'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}, if this limit exists
    • f′(c)=lim⁡x→cf(x)−f(c)x−cf'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}
    • If ff is differentiable at x=cx = c, then ff is continuous at x=cx = c.

  • Differentiation Rules

    • General and Logarithmic Differentiation Rules
      1. ddx[cu]=cu′\frac{d}{dx}[cu] = cu'
      2. ddx[u±v]=u′±v′\frac{d}{dx}[u \pm v] = u' \pm v' (sum rule)
      3. ddx[uv]=uv′+vu′\frac{d}{dx}[uv] = uv' + vu' (product rule)
      4. ddx[uv]=vu′−uv′v2\frac{d}{dx}[\frac{u}{v}] = \frac{vu' - uv'}{v^2} (quotient rule)
      5. ddx[c]=0\frac{d}{dx}[c] = 0
      6. ddx[un]=nun−1u′\frac{d}{dx}[u^n] = nu^{n-1}u' (power rule)
      7. ddx[x]=1\frac{d}{dx}[x] = 1
      8. ddx[ln⁡u]=u′u\frac{d}{dx}[\ln u] = \frac{u'}{u}
      9. ddx[eu]=euu′\frac{d}{dx}[e^u] = e^uu'
      10. ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) (chain rule)
    • Derivatives of the Trigonometric Functions
      1. ddx[sin⁡u]=(cos⁡u)u′\frac{d}{dx}[\sin u] = (\cos u)u'
      2. ddx[csc⁡u]=−(csc⁡ucot⁡u)u′\frac{d}{dx}[\csc u] = -(\csc u \cot u)u'
      3. ddx[cos⁡u]=−(sin⁡u)u′\frac{d}{dx}[\cos u] = -(\sin u)u'
      4. ddx[sec⁡u]=(sec⁡utan⁡u)u′\frac{d}{dx}[\sec u] = (\sec u \tan u)u'
      5. ddx[tan⁡u]=(sec⁡2u)u′\frac{d}{dx}[\tan u] = (\sec^2 u)u'
      6. ddx[cot⁡u]=−(csc⁡2u)u′\frac{d}{dx}[\cot u] = -(\csc^2 u)u'
    • Derivatives of the Inverse Trigonometric Functions
      1. ddx[arcsin⁡u]=u′1−u2\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1 - u^2}}
      2. ddx[arccsc u]=−u′∣u∣u2−1\frac{d}{dx}[\text{arccsc } u] = \frac{-u'}{|u|\sqrt{u^2 - 1}}
      3. ddx[arccos⁡u]=−u′1−u2\frac{d}{dx}[\arccos u] = \frac{-u'}{\sqrt{1 - u^2}}
      4. ddx[arcsec u]=u′∣u∣u2−1\frac{d}{dx}[\text{arcsec } u] = \frac{u'}{|u|\sqrt{u^2 - 1}}
      5. ddx[arctan⁡u]=u′1+u2\frac{d}{dx}[\arctan u] = \frac{u'}{1 + u^2}
      6. ddx[arccot u]=−u′1+u2\frac{d}{dx}[\text{arccot } u] = \frac{-u'}{1 + u^2}

  • Implicit Differentiation

    • Useful when you cannot easily solve for yy as a function of xx.
    • Exercise: Find dydx\frac{dy}{dx} for y3+xy−2y−x2=−2y^3 + xy - 2y - x^2 = -2
      • ddx[y3+xy−2y−x2]=ddx[−2]\frac{d}{dx}[y^3 + xy - 2y - x^2] = \frac{d}{dx}[-2]
      • 3y2dydx+(xdydx+y)−2dydx−2x=03y^2 \frac{dy}{dx} + (x \frac{dy}{dx} + y) - 2 \frac{dy}{dx} - 2x = 0
      • dydx(3y2+x−2)=2x−y\frac{dy}{dx}(3y^2 + x - 2) = 2x - y
      • dydx=2x−y3y2+x−2\frac{dy}{dx} = \frac{2x - y}{3y^2 + x - 2}

  • Higher Order Derivatives

    • Successive derivatives of f(x)f(x).
    • The second derivative of f(x)f(x), f′′(x)f''(x), is the derivative of f′(x)f'(x).
    • Numerical notation: f(n)(x)=y(n)f^{(n)}(x) = y^{(n)}
    • The second derivative is also indicated by d2ydx2\frac{d^2y}{dx^2}.
    • Exercise: Find the third derivative of y=x5y = x^5
      • y′=5x4y' = 5x^4
      • y′′=20x3y'' = 20x^3
      • y′′′=60x2y''' = 60x^2

  • Derivatives of Inverse Functions

    • If y=f(x)y = f(x) and x=f−1(y)x = f^{-1}(y) are differentiable inverse functions, then their derivatives are reciprocals: dxdy⋅dydx=1\frac{dx}{dy} \cdot \frac{dy}{dx} = 1

  • Logarithmic Differentiation

    • Useful for differentiating certain functions using logarithms.
      1. Take ln⁡\ln of both sides
      2. Differentiate
      3. Solve for y′y'
      4. Substitute for yy
      5. Simplify
    • Exercise: Find dydx\frac{dy}{dx} for y=(x2+1x2−1)1/3y = \left( \frac{x^2 + 1}{x^2 - 1} \right)^{1/3}
      • ln⁡y=13[ln⁡(x2+1)−ln⁡(x2−1)]\ln y = \frac{1}{3} [\ln(x^2 + 1) - \ln(x^2 - 1)]
      • y′y=13[2xx2+1−2xx2−1]\frac{y'}{y} = \frac{1}{3} \left[ \frac{2x}{x^2 + 1} - \frac{2x}{x^2 - 1} \right]
      • y′=(x2+1x2−1)1/3⋅13[2xx2+1−2xx2−1]y' = \left( \frac{x^2 + 1}{x^2 - 1} \right)^{1/3} \cdot \frac{1}{3} \left[ \frac{2x}{x^2 + 1} - \frac{2x}{x^2 - 1} \right]

  • Mean Value Theorem

    • If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists a number cc in (a,b)(a, b) such that f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}.

  • L'Hôpital's Rule

    • If lim⁡f(x)g(x)\lim \frac{f(x)}{g(x)} is an indeterminate form of the type 0/00/0 or ∞/∞\infty / \infty, and if lim⁡f′(x)g′(x)\lim \frac{f'(x)}{g'(x)} exists, then lim⁡f(x)g(x)=lim⁡f′(x)g′(x)\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}.
    • Indeterminate forms of 0⋅∞0 \cdot \infty can be reduced to 0/00/0 or ∞/∞\infty / \infty to apply L'Hôpital's Rule.
    • Note: L'Hôpital's Rule can be applied to the four different indeterminate forms of ∞/∞\infty / \infty: ∞/∞\infty / \infty, (−∞)/∞(-\infty) / \infty, ∞/(−∞)\infty / (-\infty), and (−∞)/(−∞)(-\infty) / (-\infty).
    • Exercise: What is lim⁡x→0sin⁡xx+1\lim_{x \to 0} \frac{\sin x}{x + 1}?
      • Answer: 1, since lim⁡x→0cos⁡x1=1\lim_{x \to 0} \frac{\cos 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.
    • Exercise: The slope of the normal line to the curve y=2x2+1y = 2x^2 + 1 at (1,3)(1, 3) is -1/4.

  • Extreme Value Theorem

    • If a function f(x)f(x) is continuous on a closed interval, then f(x)f(x) has both a maximum and minimum value in the interval.

  • Curve Sketching

    • f'(c) > 0: ff increasing at cc
    • f'(c) < 0: ff decreasing at cc
    • f′(c)=0f'(c) = 0: horizontal tangent at cc
    • f′(c)=0,f′(c−)<0,f′(c+)>0f'(c) = 0, f'(c^-) < 0, f'(c^+) > 0: relative minimum at cc
    • f'(c) = 0, f'(c^-) > 0, f'(c^+) < 0: relative maximum at cc
    • f'(c) = 0, f''(c) > 0: relative minimum at cc
    • f'(c) = 0, f''(c) < 0: relative maximum at cc
    • f′(c)=0,f′′(c)=0f'(c) = 0, f''(c) = 0: further investigation required
    • f''(c) > 0: concave upward
    • f''(c) < 0: concave downward
    • f′′(c)=0f''(c) = 0: further investigation required
    • f′′(c)=0,f′′(c−)<0,f′′(c+)>0f''(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)f(c) exists, f′(c)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−f(x<em>n)f′(x</em>n)x<em>{n + 1} = x</em>n - \frac{f(x<em>n)}{f'(x</em>n)}
    • Let x1x_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 solve practical problems requiring maximum or minimum values.
    • Exercise: A rectangular box with a square base and no top has a volume of 500 cubic inches. Find the dimensions for the box that require the least amount of material.
      • Let V=volumeV = \text{volume}, S=surface areaS = \text{surface area}, x=length of basex = \text{length of base}, and h=height of boxh = \text{height of box}
      • V=x2h=500V = x^2h = 500
      • S=x2+4xh=x2+4x(500/x2)=x2+(2000/x)S = x^2 + 4xh = x^2 + 4x(500/x^2) = x^2 + (2000/x)
      • S′=2x−(2000/x2)=0S' = 2x - (2000/x^2) = 0
      • 2x3=20002x^3 = 2000
      • x=10,h=5x = 10, h = 5
      • Dimensions: 10 x 10 x 5 inches

  • Rates-of-Change Problems

    • Distance, Velocity, and Acceleration
      • y=s(t)y = s(t): position of a particle along a line at time tt
      • v=s′(t)v = s'(t): instantaneous velocity (rate of change) at time tt
      • a=v′(t)=s′′(t)a = v'(t) = s''(t): instantaneous acceleration at time tt
    • Related Rates of Change
      • Calculus can find the rate of change of two or more variables that are functions of time tt by differentiating with respect to tt.
      • Exercise: A boy 5 feet tall walks at a rate of 3 feet/sec toward a streetlamp that is 12 feet above the ground.
        • z=125xz = \frac{12}{5} x
        • dxdt=57dydt\frac{dx}{dt} = \frac{5}{7} \frac{dy}{dt}
        • dzdt=125dxdt\frac{dz}{dt} = \frac{12}{5} \frac{dx}{dt}
        • dxdt=57(3)=157\frac{dx}{dt} = \frac{5}{7} (3) = \frac{15}{7}
        • dzdt=125(157)=367\frac{dz}{dt} = \frac{12}{5} (\frac{15}{7}) = \frac{36}{7}
      • Exercise: A conical tank 20 feet in diameter and 30 feet tall (with vertex down) leaks water at a rate of 5 cubic feet per hour. At what rate is the water level dropping when the water is 15 feet deep?
        • V=13πr2hV = \frac{1}{3} \pi r^2 h
        • dvdt=19h2dhdt\frac{dv}{dt} = \frac{1}{9} h^2 \frac{dh}{dt}
        • r=h3r = \frac{h}{3}
        • dhdt=45π2\frac{dh}{dt} = \frac{45}{\pi 2}
        • dhdt=15πft/hr\frac{dh}{dt} = \frac{1}{5 \pi} ft/hr

Integral Calculus

  • Indefinite Integrals
    • A function F(x)F(x) is the antiderivative of a function f(x)f(x) if for all xx in the domain of ff, F′(x)=f(x)F'(x) = f(x).
    • ∫f(x)dx=F(x)+C\int f(x) dx = F(x) + C, where CC is a constant.
  • Basic Integration Formulas
    • General and Logarithmic Integrals
      1. ∫kf(x)dx=k∫f(x)dx\int kf(x) dx = k \int f(x) dx
      2. ∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx
      3. ∫kdx=kx+C\int k dx = kx + C
      4. ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C, n≠−1n \neq -1
      5. ∫exdx=ex+C\int e^x dx = e^x + C
      6. ∫axdx=axln⁡a+C\int a^x dx = \frac{a^x}{\ln a} + C, a > 0, a \neq 1
      7. ∫dxx=ln⁡∣x∣+C\int \frac{dx}{x} = \ln |x| + C
    • Trigonometric Integrals
      1. ∫sin⁡xdx=−cos⁡x+C\int \sin x dx = -\cos x + C
      2. ∫cos⁡xdx=sin⁡x+C\int \cos x dx = \sin x + C
      3. ∫sec⁡2xdx=tan⁡x+C\int \sec^2 x dx = \tan x + C
      4. ∫csc⁡2xdx=−cot⁡x+C\int \csc^2 x dx = -\cot x + C
      5. ∫sec⁡xtan⁡xdx=sec⁡x+C\int \sec x \tan x dx = \sec x + C
      6. ∫csc⁡xcot⁡xdx=−csc⁡x+C\int \csc x \cot x dx = -\csc x + C
      7. ∫tan⁡xdx=−ln⁡∣cos⁡x∣+C\int \tan x dx = -\ln |\cos x| + C
      8. ∫cot⁡xdx=ln⁡∣sin⁡x∣+C\int \cot x dx = \ln |\sin x| + C
      9. ∫sec⁡xdx=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x dx = \ln |\sec x + \tan x| + C
      10. ∫csc⁡xdx=−ln⁡∣csc⁡x+cot⁡x∣+C\int \csc x dx = -\ln |\csc x + \cot x| + C
      11. ∫dxa2−x2=arcsin⁡xa+C\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C
      12. ∫dxa2+x2=1aarctan⁡xa+C\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan \frac{x}{a} + C
      13. ∫dxxx2−a2=1aarcsec xa+C\int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \text{arcsec } \frac{x}{a} + C
  • Integration by Substitution
    • ∫f(g(x))g′(x)dx=F(g(x))+C\int f(g(x))g'(x) dx = F(g(x)) + C
    • If u=g(x)u = g(x), then du=g′(x)dxdu = g'(x) dx and ∫f(u)du=F(u)+C\int f(u) du = F(u) + C
  • Integration by Parts
    • ∫udv=uv−∫vdu\int u dv = uv - \int v du
  • Distance, Velocity, and Acceleration (on Earth)
    • a(t)=s′′(t)=−32 ft/sec2a(t) = s''(t) = -32 \text{ ft/sec}^2
    • v(t)=s′(t)=∫s′′(t)dt=∫−32dt=−32t+C1v(t) = s'(t) = \int s''(t) dt = \int -32 dt = -32t + C_1
    • At t=0t = 0, v<em>0=v(0)=(−32)(0)+C</em>1=C1v<em>0 = v(0) = (-32)(0) + C</em>1 = C_1
    • s(t)=∫v(t)dt=∫(−32t+v<em>0)dt=−16t2+v</em>0t+C2s(t) = \int v(t) dt = \int (-32t + v<em>0) dt = -16t^2 + v</em>0t + C_2
  • Separable Differential Equations
    • Sometimes possible to separate variables and write a differential equation in the form f(y)dy+g(x)dx=0f(y) dy + g(x) dx = 0 by integrating: ∫f(y)dy+∫g(x)dx=C\int f(y) dy + \int g(x) dx = C
    • Exercise: Solve for dydx=−xy\frac{dy}{dx} = -\frac{x}{y}
      • 2xdx+ydy=02x dx + y dy = 0
      • x2+y22=Cx^2 + \frac{y^2}{2} = C
  • Applications to Growth and Decay
    • Often, the rate of change of a variable yy is proportional to the variable itself.
      • dydt=ky\frac{dy}{dt} = ky
      • Separate the variables: dyy=kdt\frac{dy}{y} = k dt
      • Integrate both sides: ln⁡∣y∣=kt+C1\ln |y| = kt + C_1
      • y=Cekty = 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 ff on the interval [a,b][a, b]: lim⁡<em>Δx→0∑</em>i=1nf(x<em>i)Δx=∫</em>abf(x)dx\lim<em>{\Delta x \to 0} \sum</em>{i=1}^n f(x<em>i) \Delta x = \int</em>a^b f(x) dx
  • Properties of Definite Integrals
    1. ∫<em>ab[f(x)+g(x)]dx=∫</em>abf(x)dx+∫abg(x)dx\int<em>a^b [f(x) + g(x)] dx = \int</em>a^b f(x) dx + \int_a^b g(x) dx
    2. ∫<em>abkf(x)dx=k∫</em>abf(x)dx\int<em>a^b kf(x) dx = k \int</em>a^b f(x) dx
    3. ∫aaf(x)dx=0\int_a^a f(x) dx = 0
    4. ∫<em>abf(x)dx=−∫</em>baf(x)dx\int<em>a^b f(x) dx = -\int</em>b^a f(x) dx
    5. ∫<em>abf(x)dx+∫</em>bcf(x)dx=∫acf(x)dx\int<em>a^b f(x) dx + \int</em>b^c f(x) dx = \int_a^c f(x) dx
    6. If f(x)≤g(x)f(x) \leq g(x) on [a,b][a, b], then ∫<em>abf(x)dx≤∫</em>abg(x)dx\int<em>a^b f(x) dx \leq \int</em>a^b g(x) dx
  • Approximations to the Definite Integral
    • Riemann Sums
      • ∫<em>abf(x)dx=S</em>n=∑<em>i=1nf(x</em>i)Δx\int<em>a^b f(x)dx = S</em>n = \sum<em>{i=1}^n f(x</em>i)\Delta x
    • Trapezoidal Rule
      • ∫<em>abf(x)dx≈[12f(x</em>0)+f(x<em>1)+f(x</em>2)+…+f(x<em>n−1)+12f(x</em>n)]b−an\int<em>a^b f(x)dx \approx \left[ \frac{1}{2} f(x</em>0) + f(x<em>1) + f(x</em>2) + … + f(x<em>{n-1}) + \frac{1}{2} f(x</em>n) \right] \frac{b - a}{n}
  • The Fundamental Theorem of Calculus
    • If ff is continuous on [a,b][a, b] and if F′=fF' = f, then ∫abf(x)dx=F(b)−F(a)\int_a^b f(x) dx = F(b) - F(a)
  • The Second Fundamental Theorem of Calculus
    • If ff is continuous on an open interval II containing aa, then for every xx in the interval, ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t) dt = f(x)
  • Area Under a Curve
    • If f(x)≥0f(x) \geq 0 on [a,b][a, b], then area A=∫abf(x)dxA = \int_a^b f(x) dx
    • If f(x)≤0f(x) \leq 0 on [a,b][a, b], then area A=−∫abf(x)dxA = -\int_a^b f(x) dx
    • If f(x)≥0f(x) \geq 0 on [a,c][a, c] and f(x)≤0f(x) \leq 0 on [c,b][c, b], then A=∫<em>acf(x)dx−∫</em>cbf(x)dxA = \int<em>a^c f(x) dx - \int</em>c^b f(x) dx
    • The area enclosed by the graphs of y=2x2y = 2x^2 and y=4x+6y = 4x+6 is 64/3
  • Average Value of a Function on an Interval
    • 1b−a∫abf(x)dx\frac{1}{b - a} \int_a^b f(x) dx
  • Volumes of Solids with Known Cross Sections
    1. For cross sections of area A(x)A(x), taken perpendicular to the x-axis: V=∫abA(x)dxV = \int_a^b A(x) dx
    2. For cross sections of area A(y)A(y), taken perpendicular to the y-axis: V=∫abA(y)dyV = \int_a^b A(y) dy
  • Volumes of Solids of Revolution: Disk Method
    • V=∫abπr2dxV = \int_a^b \pi r^2 dx
    • Rotated about the x-axis: V=∫abπ[f(x)]2dxV = \int_a^b \pi [f(x)]^2 dx
    • Rotated about the y-axis: V=∫abπ[f(y)]2dyV = \int_a^b \pi [f(y)]^2 dy
  • Volumes of Solids of Revolution: Washer Method
    • V=∫<em>abπ(r</em>o2−ri2)dxV = \int<em>a^b \pi (r</em>o^2 - r_i^2) dx
    • Rotated about the x-axis: V=∫<em>abπ[(f</em>1(x))2−(f2(x))2]dxV = \int<em>a^b \pi [(f</em>1(x))^2 - (f_2(x))^2] dx
    • Rotated about the y-axis: V=∫<em>abπ[(f</em>1(y))2−(f2(y))2]dyV = \int<em>a^b \pi [(f</em>1(y))^2 - (f_2(y))^2] dy
      *Volumes of Solids of Revolution Cylindrical Shell Method V = ∫ a b 2 πrh dr
      *Rotated about the x-axis: V = 2 π ∫ a b xƒ(x) dx
      *Rotated about the y-axis: V = 2 π ∫ a b yƒ(y) dy

Some Useful Formulas

  • log⁡ax=log⁡xlog⁡a\log_a x = \frac{\log x}{\log a}
  • sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  • 1+tan⁡2x=sec⁡2x1 + \tan^2 x = \sec^2 x
  • 1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x
  • sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x
  • cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x
  • sin⁡2x=12(1−cos⁡2x)\sin^2 x = \frac{1}{2}(1 - \cos 2x)
  • cos⁡2x=12(1+cos⁡2x)\cos^2 x = \frac{1}{2}(1 + \cos 2x)
  • Volume of a right circular cylinder =πr2h= \pi r^2h
  • Volume of a cone $$= \frac{1}{3} \pi r^