AP Calculus BC Practice Test

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• Let y = f(x) and f is invertible; then (f^{-1})'(y) equals what?
• Which expression is the correct quotient rule for d/dx [f(x)/g(x)]?
• What is the washers formula for volume when there is an outer radius R(x) and an inner radius r(x)?
• In solving y' - 2y = e^{3x} by integrating factor, which integrating factor μ(x) is used?
• Which statement best expresses a condition for applying L'Hôpital's Rule to lim_{x->a} f(x)/g(x)?
• Which of the following is cos(π/3)?
• An object in one-dimensional motion reverses direction when which occurs?
• Compute the average rate of change of f(x) = x^2 on the interval [1,3].
• If r(t) = (cos t, sin t), the acceleration is a(t) = (-cos t, -sin t).
• To locate absolute extrema of a function on [a,b], which points should you examine?
• Compute ∫ sqrt(9 - x^2) dx. Which expression is correct?
• For the improper integral ∫_1^∞ 1/x^p dx with p > 1, what is the value when p = 2?
• Which statement best describes partial fraction decomposition of a rational function with linear factors?
• Which condition indicates that the particle is speeding up?
• What is the interval of convergence for ∑_{n=0}^∞ (-1)^n x^n /(n+1)?
• Which statement correctly describes the relationship between displacement and total distance traveled over [a,b]?
• If f'(x) > 0 on (a,b), what can be concluded about f on (a,b)?
• For r(t) = ⟨t, t^2, t^3⟩, which expression gives the speed |v(t)|?
• What is the area enclosed by the polar curve r = 1 + cos θ for θ from 0 to 2π?
• Rolle's Theorem states that if f is continuous on [a,b] and differentiable on (a,b), then there exists a c in (a,b) such that:
• Which expression is an antiderivative of e^x with respect to x?
• If v(t) is positive, the particle is moving in which direction?
• For the polar curve r = 2 cos θ, the total arc length as θ runs from -π/2 to π/2 is?
• How do you determine the time at which a particle is at rest?
• Differentiate f(x) = sin(3x^2) with respect to x.
• Which of the following is the arc length formula for y = f(x) from a to b?
• The second derivative of the position function is the which function?
• For r(t) = ⟨t, t^2, t^3⟩, which expression represents the speed |v(t)|?
• Which integral gives the total distance traveled by a particle over the interval [a,b]?
• If F(x) = ∫_{a}^{x} f(t) dt, what is F'(x)?
• For a volume formed by discs with outer radius R(x), which formula gives the volume over [a,b]?
• Evaluate ∫_{-a}^{a} sqrt(a^2 - x^2) dx. Which value is correct?
• Which statement correctly represents the Maclaurin series for sin x truncated to degree 3?
• Where are critical values located for a function with a differentiable domain?
• What is lim_{x→0} (sin x)/x?
• If f''(a) > 0 and f'(a) = 0, what does this say about the point a on the graph?
• The fixed points of the logistic equation dP/dt = kP(1 - P/L) occur when P equals which values?
• If f is differentiable on (a,b) then f is ______ on [a,b].
• What is the radius of convergence of the Maclaurin series for ln(1+x)?
• In the Maclaurin series for e^x, what is the coefficient of x^2?
• If velocity components are x'(t) = sqrt(t) and y'(t) = t, the speed is sqrt(t + t^2).
• The derivative of velocity with respect to time is which function?
• The distance traveled along the unit circle from t=0 to t=π/2 equals which value?
• In the logistic differential equation dP/dt = kP(1 - P/L), the carrying capacity is:
• If y = sec(u) and u = x^2, dy/dx equals?
• Which condition is required for the Extreme Value Theorem to apply?
• What are the first three nonzero terms of the Maclaurin series for ln(1+x)?
• If y = sin(u), where u is a differentiable function of x, what is dy/dx?
• What does the Extreme Value Theorem guarantee for a function continuous on a closed interval [a,b]?
• Which statement correctly describes dy/dx for a parametric curve in terms of t?
• If y = a^{u(x)}, what is dy/dx?
• Which statement about absolute extrema is true?
• In the position vector r(t) = (x(t), y(t)), which statement about x(t) and y(t) is correct?
• Is z/(x+y) equal to z/x + z/y for nonzero x and y?
• If y = csc(3x), dy/dx equals?
• If f is increasing on an interval, what can be said about f' on that interval?
• What is the radius of convergence of the power series ∑_{n=1}^∞ n x^n?
• What is the derivative of sqrt(a^2 - x^2) with respect to x?
• The acceleration vector of a particle with position r(t) = (x(t), y(t)) is which of the following?
• Using Taylor expansion to bound the error of approximating e^1 by the 4-term polynomial P3(1). Which inequality correctly bounds the remainder R3?
• If y = sin(3x), dy/dx = ?
• Using the Fundamental Theorem of Calculus, what is ∫_0^2 4x dx?
• If y = tan(2x + 1), dy/dx = ?
• If y = sec(3x), dy/dx = ?
• Which is the standard antiderivative of sqrt(a^2 - x^2) dx?
• An inflection point occurs where the second derivative is zero or undefined and a sign change occurs in which of the following?
• The distance traveled by a particle along a parametric path r(t) from t = a to t = b is given by which integral?
• Which derivative corresponds to F(x) = -4 ln|x| + 7 ln|x-1|?
• If lim_{x→c} f(x) exists and equals L, and f(c) = L, what does this imply about f at c?
• With V = (1/12) π h^3, if h' = 3 cm/s and h = 8 cm, what is dV/dt?
• The second derivative of a parametric curve is given by which formula?
• For the parametric curves x = sin t, y = cos t, dy/dx simplifies to which expression?
• Which statement is true about Rolle's Theorem?
• The second derivative provides information about which property of a function?
• What is d/dx [sin(5x^2)]?
• If y = csc(u), dy/dx equals?
• A 10-foot ladder leaning against a wall. The top slides down with dy/dt = -2 ft/s when the top is at height y = 6 ft. How fast is the bottom moving away from the wall when y = 6 ft?
• Which statement describes the Intermediate Value Theorem?
• Which condition identifies a location of a relative maximum for a differentiable function?
• Which statement is true about the Maclaurin series?
• Compute ∫ cos x dx.
• The instantaneous rate of change at x for a function f is represented by which expression?
• In a Taylor series expansion about a, what is the coefficient of (x-a)^2 in f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ...?
• If y = ln(u(x)), dy/dx = ?
• Compute ∫ x e^x dx by parts; which expression represents its antiderivative?
• Compute ∫_0^{π/2} sin^2 x dx.
• What is the average value of f on [0,2] for f(x) = x^2?
• Let F(x) = ∫ from a to u(x) f(t) dt, where u is differentiable. Then F'(x) equals what?
• If y = cos(u), where u is a differentiable function of x, what is dy/dx?
• In the logistic model, the population grows fastest when P equals which value?
• A function f has average value on [0,2] equal to (1/2) ∫_0^2 f(x) dx. What is the average value of f(x) = x^2 on [0,2]?
• Which expression represents the Maclaurin polynomial of degree 3 for e^x?
• Which statement correctly describes continuity at a point?
• Compute the arc length of the parametric curve x = t^2, y = t^3 from t = 0 to t = 1.
• Which expression is an antiderivative of e^x?
• Let F(x) = ∫_{0}^{x} t^2 dt. What is F'(x)?
• If f(1) = 3 and f(5) = 11, there exists c in (1,5) with f'(c) equal to (f(5) - f(1))/(5 - 1). What is that value?
• Which of the following is the correct integration by parts formula?
• If f'(x) = x^3 - 3x, what is f'(2)?
• Which method is used to approximate the definite integral ∫ f(x) dx over an interval?
• Evaluate lim_{x→0} (1 - cos x)/x^2.
• Using the ratio test, the series ∑ n! / n^n converges because the limit of the ratio a_{n+1}/a_n equals which value?
• Which expression represents the formal derivative at a point x using the limit definition?
• For the parametric curve x = t^2, y = t^3, dy/dx expressed in terms of t is which expression?
• Evaluate lim_{x→∞} (1 + 1/x)^x.
• The antiderivative of 2x cos(x^2) dx is which expression?
• The instantaneous speed of a particle moving with position r(t) = (x(t), y(t)) is given by which expression?
• Which statement defines a critical point?
• For the series ∑ x^n /(n+1), what is its interval of convergence?
• Which expression represents the Maclaurin series for cos x truncated to degree 2?
• The distance traveled along the unit circle from t=0 to t=π equals which value?
• The derivative of the position function is the ______ function.
• In the disc method, the radius R represents:
• Compute ∫ sin(x) dx.
• Does ∫_1^∞ 1/x^2 dx converge, and if so, what is its value?
• Which expression correctly represents the volume of a solid formed by cross sections perpendicular to the x-axis with cross-sectional area A(x)?
• If a rational function has a canceled factor producing a hole, how does this affect vertical asymptotes?
• If y = cot(u), dy/dx equals?
• For the parametric curve x = cos t, y = sin t, what is dy/dx in terms of t?
• If u is a differentiable function of x, what is d/dx of e^{u(x)}?
• Mean Value Theorem requires f to be continuous on [a,b] and differentiable on (a,b). What does it guarantee?
• Which formula correctly gives the derivative of the inverse function (f^{-1})'(y)?
• A rectangle in the first quadrant under y = 9 - x^2 has area A = x(9 - x^2). Find x that maximizes A.
• Which of the following equals sin(π/6)?
• Compute lim_{x->0} sin x / x.
• The arc length of a parametric curve x(t), y(t) from t=a to t=b is computed by which integral?
• Does the harmonic series ∑ 1/n diverge or converge?
• The velocity vector for a particle with position r(t) = (x(t), y(t)) is what?
• How do you find the displacement of a particle over the interval [a,b]?
• Which statement correctly represents the Maclaurin series for cos x truncated to degree 2?
• Which expression represents the Maclaurin series for sin x truncated to degree 3?
• If a function is concave up on an interval, what does that say about its second derivative on that interval?
• What is the formula for the speed of a function?
• For the circle x^2 + y^2 = 25, dy/dx equals -x/y. What is dy/dx at the point (3,4)?
• Is (x+y)/z equal to x/z + y/z for nonzero z?
• What is the formula for the area between two curves y = f(x) and y = g(x) on [a,b] when f is above g?
• If y = a^x, where a > 0 and a ≠ 1, what is dy/dx?
• Compute ∫ a^x dx for a > 0, a ≠ 1.
• A cone with height h(t) increasing; if h' = 2 cm/s, H=10, R=5, find dV/dt when h=6 for V = (1/12)π h^3.
• Which option best lists the four columns in the Euler's Method chart for a first-order ODE?
• The derivative of f(g(x)) with respect to x is?
• For the unit circle path r(t) = (cos t, sin t), the distance traveled from t=0 to t=π is represented by which integral?
• In partial fraction decomposition, to find a constant a in a term a/(...) you typically do what?
• For a parametric curve x(t), y(t), the slope dy/dx at a given t is given by which expression?
• Does the p-series ∑ 1/n^p converge for p>1?
• Which of the following is a correct form of the volume by washers formula for outer radius R(x) and inner radius r(x)?
• Using a centered difference with h = 0.1, approximate f'(2) for f(x) = x^2.
• Evaluate ∫_0^a sqrt(a^2 - x^2) dx. Which value is correct in terms of a?
• Using the method of shells, what is the volume of the solid formed by rotating y = sqrt(x) from x = 0 to 4 about the y-axis?
• For a rational function f(x)/g(x), which statement correctly describes its horizontal asymptote based on the degrees of the polynomials?
• When approximating f'(x) numerically at a point, which concept is typically used?
• For the polar curve r = 1 + cos θ, the area enclosed by the curve as θ runs from 0 to 2π is what value?
• The speed of a particle with velocity v(t) = (dx/dt, dy/dt) is which expression?
• A point c is a relative minimum if which condition holds?
• Which condition is not required for the Mean Value Theorem?
• If x'(t) = 3t and y'(t) = 2, the speed is which expression?
• What is lim_{n→∞} a_n for a_n = n/(n+1)?
• Which expression correctly represents the nth-degree Taylor polynomial of f about a?
• Evaluate lim_{x→0} ln(1+x)/x.
• Compute the dot product v·w for v = ⟨1, -1, 2⟩ and w = ⟨-2, 0, 3⟩.
• Using the Maclaurin series for e^x truncated after the x^3 term, approximate e^1.
• What is the radius of convergence for ∑ (-1)^n x^n /(n+1)?
• If f''(x) > 0 on (a,b), the function is concave up on that interval. Which choice best expresses this?
• Which value equals sin(π/4)?
• Using washers, find the volume of the solid formed by rotating y = x^2 from x = 0 to 2 about the x-axis.
• To locate a vertical asymptote of a rational function, which procedure is described?
• If v(t) is negative, the particle is moving in which direction?
• Which of the following is a correct antiderivative form for ∫ sqrt(1+4x^2) dx used in arc length?
• What is the coefficient of x^4 in the Maclaurin series for e^x?
• If velocity components are x'(t) = t and y'(t) = t^2, the speed is sqrt(t^2 + t^4).
• If f'(x) = x^2 and f(0) = 4, what is f(2)?
• Using a forward difference, approximate f'(2) for f(x) = x^2 with h = 0.1.
• If a function has cross-sectional area A(x) perpendicular to the x-axis, its volume over [a,b] is:
• Find the arc length of y = x^2 from x = 0 to x = 1.
• What is the coefficient of x^3 in the Maclaurin series for ln(1+x)?
• Solve the differential equation y' - 2y = e^{3x} by integrating factors. Which is the general solution?
• Which of the following is NOT a condition for f to be continuous at c?
• Which statement best indicates the particle is slowing down?
• If y = tan(u), dy/dx equals?
• According to the Mean Value Theorem, there exists c in (0,4) with f'(c) equal to the average rate of change of f on [0,4]. If f(0) = 2 and f(4) = 18, what is f'(c)?
• For f'(x) = x^3 - 3x, on which intervals is f increasing?
• The alternating harmonic series ∑ (-1)^n / n converges. Does it converge absolutely, conditionally, or diverge?
• Which expression is the product rule for d/dx [f(x) g(x)]?
• If a > 0, what is the derivative of arcsin(x/a) with respect to x?
• Which growth order correctly ranks exponential, polynomial, and logarithmic functions as x grows large?
• What is the average value of f on [0, π] when f(x) = sin x?
• If x^2 + y^2 = 4, what is dy/dx at the point (√3, 1)?
• Which antiderivative corresponds to ∫ (3x+4)/(x^2 - x) dx?
• Evaluate dy/dx at t = π/4 for x = cos t, y = sin t.
• Which method is typically used to find the volume of a solid of revolution around the y-axis using vertical slices?
• Which approach helps organize potential extrema by listing end points and critical points and comparing their y-values?
• The sum of the geometric series ∑_{n=0}^∞ (1/2)^n equals how much?
• In a logistic model, the carrying capacity L is the maximum sustainable population.
• The acceleration is the derivative of the velocity.
• If y = sec(u), dy/dx equals?