BC Calculus Practice Test with Answers
Sample 1

A collection of BC Calculus practice questions, complete with answers, similar to those found on the AP Calculus exam. Solutions are provided at the bottom of the page.

1. If \[ \frac{dy}{dx} = (2x - 1)y \], and \( y(1) = e \), find \( y(2) \).
   A) \( 3e \)
   B) \( 3^{e} \)
   C) \( \frac{e}{3} \)
   D) \( \frac{3}{e} \)
   E) \( e^{3} \)
2. A particle moves along the x-axis with velocity \( v(t) = 2t^{2} - t + 1 \). When \( t = 0 \), the particle is at \( x = -3 \). Find the particle's position at \( t = 2 \).
   A) \( 7 \)
   B) \( \frac{7}{3} \)
   C) \( 1 \)
   D) \( 22 \)
   E) \( \frac{3}{7} \)
3. Evaluate the integral: \[ \int_{3}^{4} \frac{5x + 5}{x^{2} + x - 6} \, dx \]
   A) \( 0 \)
   B) \( \ln\left(\frac{7}{3}\right) + \ln(2) \)
   C) \( 2\ln\left(\frac{7}{3}\right) - \ln(2) \)
   D) \( 2\ln\left(\frac{7}{3}\right) + \ln(2) \)
   E) \( 3\ln\left(\frac{7}{3}\right) \)
4. Evaluate the integral: \[ \int \frac{1}{x(1 + (\ln x)^{2})} \, dx \]
   A) \( \arctan(\ln x) + C \)
   B) \( \frac{1}{2}x^{2}(\ln^{2}(x) \ln(x)) + \frac{3}{4}x^{2} + C \)
   C) \( \left[ \frac{1}{2}x^{2}(\ln^{2}(x) \ln(x)) + \frac{3}{4}x^{2} \right]^{-1} + C \)
   D) \( \arcsin(\ln x) + C \)
   E) \( \frac{1}{\arctan(\ln x)} + C \)
5. If \( f(x) = 2 + \ln(x + 3) \), then its inverse \( f^{-1}(x) \) is:
   A) \( f^{-1}(x) = (2 + \ln(x + 3))^{-1} \)
   B) \( f^{-1}(x) = \ln(x - 2) \)
   C) \( f^{-1}(x) = -(2 + \ln(x + 3))^{-1} \)
   D) \( f^{-1}(x) = e^{x - 2} - 3 \)
   E) \( f^{-1}(x) = e^{x + 2} - 3 \)
6. If \( f'(x) < 0 \) and \( f''(x) < 0 \) for all \( x \), which graph represents \( f \)?
   
7. The set of all \( K \) such that \( f(x) = x^{4} - 14x^{2} + 24x + K \) has two distinct x-intercepts is:
   A) \( (0, \infty) \)
   B) \( (-\infty, \infty) \)
   C) \( (-100, -11) \cup (-11, 130) \)
   D) \( (-8, 127) \)
   E) \( (-\infty, -11) \cup (-8, 117) \)
8. If \( f(-x) = f(x) \) and \( f \) is differentiable for all \( x \), which must be true?
   A) \( f'(-x) = \frac{1}{f'(x)} \)
   B) \( f'(-x) = f'(x) \)
   C) \( f'(-x) = -f'(x) \)
   D) \( f'(-x) = -\frac{1}{f'(x)} \)
   E) \( f'(-x) = (f'(x))^{-1} \)
9. If \( y = x^{x + 1} \), then \( y' = \)
   A) \( (x + 1)x^{x} \)
   B) \( (x + 1)x^{x - 1} \)
   C) \( x \ln(x^{x}) + x + 1 \)
   D) \( x^{x}(x \ln x + x + 1) \)
   E) \( x^{x} \cdot x \ln x \)
10. If \( x = \ln(t + 1) \) and \( y = \ln(t + 2) \), then \( \frac{dy}{dx} = \)
    A) \( 1 \)
    B) \( \frac{t + 1}{t + 2} \)
    C) \( \frac{t + 2}{t + 1} \)
    D) \( \frac{1}{(t + 2)\ln(t + 1)} - \frac{\ln(t + 2)}{(t + 1)\ln^{2}(t + 1)} \)
    E) \( \frac{1}{t + 2} \)
11. \[ \lim_{x \to 0} \frac{\sin(x)-\sin(2x)}{x} \]
    A) \( 1 \)
    B) \( 2 \)
    C) \( 0 \)
    D) \( -2 \)
    E) \( -1 \)
12. If \( y = \sin(\sin(\sin(x))) \), then \( \frac{dy}{dx} = \)
    A) \( \cos(\sin(\sin(x))) \)
    B) \( \cos(\cos(\sin(x))) \)
    C) \( \cos(x) \cos(\sin(x)) \cos(\sin(\sin(x))) \)
    D) \( \cos(\cos(\cos(x))) \)
    E) \( \cos(x) \cos(\sin(x)) \cos(\cos(\sin(x))) \)
13. Find \( \frac{dy}{dx} \) if \( x = \ln(y - e^{-y}) \).
    A) \( y + e^{-y} \)
    B) \( \frac{y - e^{-y}}{1 - e^{-y}} \)
    C) \( \frac{y + e^{-y}}{1 - e^{-y}} \)
    D) \( \frac{y - e^{-y}}{1 + e^{-y}} \)
    E) \( y - e^{-y} \)
14. Find \( \frac{dy}{dx} \) for \( x = \ln t + t \) and \( y = t - \ln t \).
    A) \( \frac{t - 1}{t + 1} \)
    B) \( \frac{t + 1}{t - 1} \)
    C) \( -\frac{t - 1}{t + 1} \)
    D) \( -\frac{t + 1}{t - 1} \)
    E) \( 1 \)
15. Given \( g(x) = f(h(x)) \), \( f(0) = 1 \), \( h(0) = 2 \), \( g'(0) = 3 \), \( f'(0) = 7 \), \( f'(2) = 6 \), and \( f'(5) = 8 \), find \( h'(0) \).
    A) \( 2 \)
    B) \( \frac{1}{2} \)
    C) \( -\frac{1}{2} \)
    D) \( 1 \)
    E) \( -1 \)
16. If \( f(g(x)) = 2x \) and \( f(t) = e^{2t + 1} \), then \( g(x) = \)
    A) \( \ln(2x) \)
    B) \( e^{2x} \)
    C) \( \ln(2x) + 1 \)
    D) \( \frac{\ln(2x) - 1}{2} \)
    E) \( \ln(x) \)
17. Half a period of \( y = \sin x \) from \( 0 \) to \( \pi \) is split into two equal areas by a line through the origin intersecting the curve at \( x = K \). Which equation does \( K \) satisfy?
    
    A) \( \sin K = K \)
    B) \( \cos K = K \sin K \)
    C) \( K \cos K = \sin K \)
    D) \( (2 + K)\sin K + 2\cos K = 0 \)
    E) \( K \sin K + 2\cos K = 0 \)
18. Which graph represents the polar curve \( r = -\csc t \)?
    
19. Find \( \frac{dy}{dx} \) if \( y = |x^{2} + 2x - 1| \).
    A) \( \frac{(2x + 2)(x^{2} + 2x - 1)}{|x^{2} + 2x - 1|} \)
    B) \( |2x + 2| \)
    C) \( -|2x + 2| \)
    D) \( |2x - 2| \)
    E) \( -2x + 2 \)
20. For \( f(x) = x^{5} + 5x^{4} - 40x^{2} - 80x - 48 \), which statement is true?
    A) It has no x-intercepts
    B) It has 3 inflection points
    C) It is concave up for all \( x \)
    D) It has 1 inflection point
    E) It has 2 inflection points

Answers

1. E
2. B
3. D
4. A
5. D
6. C
7. E
8. C
9. D
10. B
11. E
12. C
13. D
14. A
15. B
16. D
17. E
18. E
19. A
20. D

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