AP-Calculus-AB-Study-Guide

AP Calculus AB Exam Study Guide Notes

Key Exam Details

  • The AP Calculus AB exam is a 3-hour and 15-minute end-of-course test.
  • It consists of 45 multiple-choice questions (50% of the exam) and 6 free-response questions (50% of the exam).
  • The exam covers the following content categories:
    • Limits and Continuity: 10–12%
    • Differentiation: Definition and Basic Derivative Rules: 10–12%
    • Differentiation: Composite, Implicit, and Inverse Functions: 9–13%
    • Contextual Applications of Differentiation: 10–15%
    • Applying Derivatives to Analyze Functions: 15–18%
    • Integration and Accumulation of Change: 17–20%
    • Differential Equations: 6–12%
    • Applications of Integration: 10–15%

Limits and Continuity

  • Around 10-12% of the AP Calculus AB exam questions will be on Limits and Continuity.
  • Limits: The limit of a function ff as xx approaches cc is LL if the value of ff can be made arbitrarily close to LL by taking xx sufficiently close to cc (but not equal to cc).
  • If such a value exists, this is denoted as lim⁡x→cf(x)=L\lim_{x \to c} f(x) = L. If no such value exists, we say that the limit does not exist, abbreviated DNE.
  • Limits can be found using tables, graphs, and algebra.
  • Example: Based on the table values provided, it appears that lim⁡x→1f(x)=2.5\lim_{x \to 1} f(x) = 2.5.
  • Important algebraic techniques for finding limits include factoring and rationalizing radical expressions.
  • Suppose lim⁡<em>x→cf(x)=L\lim<em>{x \to c} f(x) = L, lim⁡</em>x→cg(x)=M\lim</em>{x \to c} g(x) = M, and lim⁡x→ch(x)=N\lim_{x \to c} h(x) = N.
    • lim⁡x→c[f(x)+g(x)]=L+M\lim_{x \to c} [f(x) + g(x)] = L + M
    • lim⁡x→c[f(x)−g(x)]=L−M\lim_{x \to c} [f(x) - g(x)] = L - M
    • lim⁡x→c[af(x)]=aL\lim_{x \to c} [af(x)] = aL , where aa is any real number.
    • lim⁡x→cf(x)g(x)=LM\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}, as long as M≠0M \neq 0
    • lim⁡x→cNh(f(x))=N\lim_{x \to c} N^{h(f(x))} = N
  • For many common functions (polynomial, rational, exponential, logarithmic, and trigonometric functions), evaluating limits requires evaluating the function at the point cc (assuming the function is defined at the point).
  • Two special limits important in calculus are:
    • lim⁡x→0sinxx=1\lim_{x \to 0} \frac{sin x}{x} = 1
    • lim⁡x→01−cosxx=0\lim_{x \to 0} \frac{1 - cos x}{x} = 0
  • One-Sided Limits: Sometimes we are interested in the value that a function ff approaches as xx approaches cc from only a single direction.
    • If the values of ff get arbitrarily close to LL as xx approaches cc while taking on values greater than cc, we say lim⁡x→c+f(x)=L\lim_{x \to c^+} f(x) = L.
    • Similarly, if xx is taking on values less than cc, we write lim⁡x→c−f(x)=L\lim_{x \to c^-} f(x) = L.
  • lim⁡<em>x→cf(x)\lim<em>{x \to c} f(x) exists if and only if both lim⁡</em>x→c+f(x)\lim</em>{x \to c^+} f(x) and lim⁡x→c−f(x)\lim_{x \to c^-} f(x) exist and have the same value.
  • A limit can fail to exist in a few ways:
    • lim⁡x→c+f(x)\lim_{x \to c^+} f(x) does not exist.
    • lim⁡x→c−f(x)\lim_{x \to c^-} f(x) does not exist.
    • Both of these one-sided limits exist, but have different values.
  • Infinite Limits, Limits at Infinity, and Asymptotes
    • When a function has a vertical asymptote at x=cx = c, the behavior of the function can be described using infinite limits.
    • If the function values increase as they approach the asymptote, we say the limit is ∞\infty, whereas if the values decrease as they approach the asymptote, the limit is −∞-\infty.
    • Saying that a limit is ±∞\pm \infty is a convenient way to describe the behavior of the function approaching the point.
    • We can also extend limits by considering how the function behaves as x→±∞x \to \pm \infty.
    • If lim⁡x→±∞f(x)=L\lim_{x \to \pm \infty} f(x) = L, then ff has a horizontal asymptote y=Ly = L. It is possible for a function to have two horizontal asymptotes, since it can have different limits as x→∞x \to \infty and x→−∞x \to -\infty.
  • The Squeeze Theorem
    • If the graph of a function lies between the graphs of two other functions, and if the two other functions share a limit at a certain point, then the function in between also shares that same limit.
    • Formally, if f(x)≤g(x)≤h(x)f(x) \le g(x) \le h(x) for all xx in some interval containing cc, and if lim⁡<em>x→cf(x)=lim⁡</em>x→ch(x)=L\lim<em>{x \to c} f(x) = \lim</em>{x \to c} h(x) = L, then lim⁡x→cg(x)=L\lim_{x \to c} g(x) = L as well.
    • Example: Since −1≤sin(x)≤1-1 \le sin(x) \le 1 for all real numbers xx, −x22≤sin(x)x≤x22\frac{-x^2}{2} \le \frac{sin(x)}{x} \le \frac{x^2}{2}. Since both end functions limit to 0, lim⁡x→0sin(x)x=0\lim_{x \to 0} \frac{sin(x)}{x} = 0.
  • Continuity: The function ff is said to be continuous at the point x=cx = c if it meets the following criteria:
    1. lim⁡x→cf(x)\lim_{x \to c} f(x) exists
    2. f(c)f(c) exists
    3. lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)
    • Each of the previously mentioned criteria can fail, resulting in a discontinuity at x=cx = c.
    • The discontinuity in graph B is referred to as a jump discontinuity.
    • In graph C, the discontinuity could be fixed by moving a single point; it occurs whenever the second condition above is satisfied and is called a removable discontinuity. If lim⁡x→cf(x)\lim_{x \to c} f(x) exists, but ff has a discontinuity at x=cx = c because it fails one of the other conditions, the discontinuity can be removed by defining or redefining f(c)f(c) to be equal to the limit at that point.
  • A function is continuous on an interval if it is continuous at every point in the interval. The following categories of functions are continuous at every point in their respective domains:
    • Polynomial
    • Rational
    • Power
    • Exponential
    • Logarithmic
    • Trigonometric
  • If ff is a piecewise-defined function with continuous component functions, then checking for continuity consists of checking whether it is continuous at its boundary points. Continuity at a boundary point requires that the functions on both sides of the point give the same result when evaluated at the point.
  • Intermediate Value Theorem: The Intermediate Value Theorem applies to continuous functions on an interval [a,b][a, b]. If dd is any value between f(a)f(a) and f(b)f(b), then there must be at least one number cc between aa and bb such that f(c)=df(c) = d.

Differentiation: Definition and Fundamental Properties

  • On the AP exam, 10–12% of questions will cover Differentiation: Definition and Fundamental Properties.
  • Definition of the Derivative
    • The average rate of change of a function ff over the interval from x=ax = a to x=a+hx = a + h is f(a+h)−f(a)h\frac{f(a + h) - f(a)}{h}.
    • Alternatively, if x=a+hx = a + h, this can be written as f(x)−f(a)x−a\frac{f(x) - f(a)}{x - a}.
  • When hh approaches 0, the limit that results is called the instantaneous rate of change of ff at x=ax = a, or the derivative of ff at x=ax = a, and is denoted f′(a)f'(a).
    • f′(a)=lim⁡<em>h→0f(a+h)−f(a)hf'(a) = \lim<em>{h \to 0} \frac{f(a + h) - f(a)}{h}, or equivalently, f′(a)=lim⁡</em>x→af(x)−f(a)x−af'(a) = \lim</em>{x \to a} \frac{f(x) - f(a)}{x - a}.
    • If this limit exists, ff is said to be differentiable at aa.
    • Graphically, f′(a)f'(a) represents the slope of the line tangent to the graph of f(x)f(x) at the point where x=ax = a.
    • Therefore, the line tangent to f(x)f(x) at x=ax = a is y−f(a)=f′(a)(x−a)y - f(a) = f'(a) (x - a).
    • If the function ff is differentiable at all points in some interval, we can define a new function on that interval by finding the derivative at every point. This new function, called the derivative of ff, can be denoted f′(x)f'(x), dydx\frac{dy}{dx}, or y′y', and is defined by f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.
    • The value of the derivative at a particular point, x=ax = a, can then be denoted f′(a)f'(a) or dydx∣x=a\frac{dy}{dx}|_{x = a}.
    • If ff is differentiable at x=ax = a, then it also must be continuous at x=ax = a.
    • Another way that differentiability can fail is via the presence of sharp turns or cusps in a graph.
  • Basic Derivatives and Rules
    • The constant rule: ddxc=0\frac{d}{dx} c = 0
    • The power rule: ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}, for any real number nn
    • The sum rule: ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)
    • The difference rule: ddx[f(x)−g(x)]=f′(x)−g′(x)\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)
    • The constant multiple rule: ddx[cf(x)]=cf′(x)\frac{d}{dx} [cf(x)] = cf'(x)
    • The product rule: ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
    • The quotient rule: ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
    • ddxcx=1\frac{d}{dx} c x = 1 and ddxx=1\frac{d}{dx} x = 1
    • Derivatives of some common functions:
      • ddxex=ex\frac{d}{dx} e^x = e^x
      • ddxlnx=1x\frac{d}{dx} ln x = \frac{1}{x}
      • ddxsinx=cosx\frac{d}{dx} sin x = cos x
      • ddxcosx=−sinx\frac{d}{dx} cos x = -sin x
      • ddxtanx=sec2x\frac{d}{dx} tan x = sec^2 x
      • ddxcscx=−cscxcotx\frac{d}{dx} csc x = -csc x cot x
      • ddxsecx=secxtanx\frac{d}{dx} sec x = sec x tan x
      • ddxcotx=−csc2x\frac{d}{dx} cot x = -csc^2 x

Differentiation: Composite, Implicit, and Inverse Functions

  • Between 9–13% of the questions on AP Calculus exam will cover Differentiation: Composite, Implicit, and Inverse Functions.
  • Chain Rule
    • The chain rule makes it possible to differentiate composite functions.
    • If y=f(g(x))y = f(g(x)), then the chain rule states that y′=f′(g(x))⋅g′(x)y' = f'(g(x)) \cdot g'(x). In alternative notation, if y=f(u)y = f(u) and u=g(x)u = g(x), then dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.
    • The chain rule can be extended to compositions of more than two functions by considering that g(x)g(x), as described previously, may itself be a composition. If y=f(g(h(x)))y = f(g(h(x))), two applications of the chain rule yield y′=f′(g(h(x)))⋅g′(h(x))⋅h′(x)y' = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x).
  • Implicit Differentiation and Inverse Functions
    • A function may sometimes be presented in implicit, rather than explicit, form. That is, it may not be given as y=f(x)y = f(x), but rather as an equation that relates xx and yy to each other. In such cases, we say that yy is implicitly defined as a function of xx.
    • Implicit differentiation is the process of finding the derivative dydx\frac{dy}{dx} for such functions, and it is accomplished by applying the chain rule.
  • Consider the equation x3+y3+xy=5x^3 + y^3 + xy = 5. Differentiating both sides of the equation with respect to xx, and remembering that we are assuming that yy is, in fact, a function of xx (so that the chain rule applies), we get the following: 3x2+3y2dydx+y+xdydx=03x^2 + 3y^2 \frac{dy}{dx} + y + x \frac{dy}{dx} = 0
  • This technique can also be applied to find the derivatives of inverse functions. Consider an invertible function ff, with inverse f−1f^{-1}. By definition, this means that f(f−1(x))=xf(f^{-1}(x)) = x. Now, differentiating both sides with respect to xx, we get f′(f−1(x))⋅(f−1)′(x)=1f'(f^{-1}(x)) \cdot (f^{-1})'(x) = 1. Solving for (f−1)′(x)(f^{-1})'(x), we have (f−1)′(x)=1f′(f−1(x))(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}.
  • Higher Order Derivatives
    • The derivative f′f' of a function is itself a function that may be differentiable. If it is, then its derivative is f′′f'', called the second derivative of f. The relationship of f′f' and f′′f'' is identical to the relationship between f and f′f'. Similarly, the derivative of f′f' is f′′′f''', the third derivative of f. This process can continue indefinitely, as long as the functions obtained continue to be differentiable. After three, the notation changes, so that the 4th derivative of f is denoted f(4)f^{(4)}, and the nth derivative is f(n)f^{(n)}.

Contextual Applications of Differentiation

  • About 10–15% of questions on the exam will cover Contextual Applications of Differentiation.
  • In any context, the derivative of a function can be interpreted as the instantaneous rate of change of the independent variable with respect to the dependent variable. If y=f(x)y = f(x), then the units of the derivative are the units of yy divided by the units of xx.
  • Straight-Line Motion
    • Rectilinear (straight-line) motion is described by a function and its derivatives.
    • If the function s(t)s(t) represents the position along a line of a particle at time tt, then the velocity is given by v(t)=s′(t)v(t) = s'(t). When the velocity is positive, the particle is moving to the right; when it is negative, the particle is moving to the left. The speed of the particle does not take direction into account, so it is the absolute value of the velocity, or ∣v(t)∣|v(t)|.
    • The acceleration of the particle is a(t)=v′(t)=s′′(t)a(t) = v'(t) = s''(t). The velocity is increasing when a(t)a(t) is positive and decreasing when a(t)a(t) is negative. The speed, however, is only increasing when v(t)v(t) and a(t)a(t) have the same sign (positive or negative). When v(t)v(t) and a(t)a(t) have different signs, the particle’s speed is decreasing.
  • Related Rates
    • Related rates problems involve multiple quantities that are changing in relation to each other. Derivatives, and especially the chain rule, are used to solve these problems.
    • Though the problems vary widely with context, there are a few steps that usually lead to a solution.
      1. Draw a picture and label relevant quantities with variables.
      2. Express any rates of change given in the problem as derivatives.
      3. Express the rate of change you need to solve for as a derivative.
      4. Relate the variables involved in the rates of change to each other with an equation.
      5. Differentiate both sides of the equation with respect to time. This may involve applying many derivative rules but will always involve the chain rule.
      6. Substitute all of the given information into the resulting equation.
      7. Solve for the unknown rate.
  • Linearization
    • The line tangent to a function at x=cx = c is the best possible linear approximation to the function near x=cx = c. Because of this, the tangent line, seen as a function L(x)L(x), is also called the linearization of the function at the given point.
  • L'Hospital's Rule
    • When two functions f and g either both have limits of 0 or both have infinite limits, we say that the limit of their ratio is an indeterminate form, represented by 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}. Limits that result in one of these forms can be evaluated using L’Hospital’s rule.
    • The full statement of L’Hospitals rule is as follows: if lim⁡<em>x→cf(x)g(x)\lim<em>{x \to c} \frac{f(x)}{g(x)} approaches 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, then lim⁡</em>x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim</em>{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.
    • In other words, when we encounter one of these indeterminate forms, we can take the derivative of each of the functions, and then reevaluate the limit.

Analytical Applications of Differentiation

  • Anywhere from 15-18% of the questions on your AP exam will cover Analytical Applications of Differentiation.
  • Mean Value Theorem
    • The Mean Value Theorem states that if ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there is at least one point between a and b at which the instantaneous rate of change of ff is equal to its average range of change over the entire interval. In other words, there is at least one value c in the interval (a,b)(a, b) for which f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}.
  • Intervals of Increase and Decrease and the First Derivative Test
    • When the derivative of a function is positive, the function increases, and when the derivative is negative, the function decreases. To find intervals on which a function is increasing or decreasing, then, it is necessary to solve for where its derivative is positive or negative. The procedure for doing this involves first finding the values, called critical points, at which the derivative is zero or undefined.
    • If ff changes from increasing to decreasing at x=cx = c, ff has a local maximum at cc. If it changes from decreasing to increasing at x=cx = c, it has a local minimum at cc. Taken together, local maximums and local minimums are referred to as local extrema.
    • The first derivative test summarizes these facts and describes the process of finding local maximums and minimums. Specifically, suppose x=cx = c is a critical point of ff. Then:
      • if f′f' is positive to the left of cc, and negative to the right of cc, then ff has a local maximum at cc.
      • if f′f' is negative to the left of cc, and positive to the right of cc, then ff has a local minimum at cc.
      • if neither of the above conditions apply, ff does not have a local extreme at cc.
  • Absolute Extrema
    • If MM is the largest value that ff attains on some interval II containing cc, then MM is called the global maximum of ff on II. Similarly, if MM is the smallest value that ff attains on some interval II containing cc, then MM is called the global minimum of ff on II.
    • There is no reason to expect that an arbitrary function has a global maximum or minimum value on a given interval. However, the Extreme Value Theorem guarantees that a function does have a global maximum and a global minimum on any closed interval on which it is continuous. On such an interval, both of the global extrema must occur at either a critical point or at an endpoint of the interval.
    • The Candidate Test gives a procedure for finding these global extrema on a closed interval [a,b][a, b]:
      1. Check that ff is continuous on [a,b][a, b]