Free AP Calculus BC Practice Exam (PDF)

This free AP Calculus BC practice exam concentrates on the material that separates BC from AB: Taylor and Maclaurin series with the alternating-series and Lagrange error bounds, interval of convergence by the ratio test, parametric arc length, polar area, vector-valued motion, integration by parts and partial fractions, improper integrals, Euler's method and the logistic equation. Students often lose BC points on the endpoints of an interval of convergence, on the missing one-half in a polar area integral, and on error bounds stated without a justification. Every solution shows the working graders look for.

1. Question 12 marks · Limits (L'Hôpital)
   Evaluate $\displaystyle\lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{x^2}$.
   • A.
     $0$
   • B.
     $1$
   • C.
     $2$
   • D.
     $4$
2. Question 22 marks · Integration techniques
   $\displaystyle\int x e^{3x}\,dx =$
   • A.
     $\dfrac{e^{3x}}{9}(3x - 1) + C$
   • B.
     $\dfrac{e^{3x}}{9}(3x + 1) + C$
   • C.
     $\dfrac{x e^{3x}}{3} + C$
   • D.
     $\dfrac{x^2 e^{3x}}{6} + C$
3. Question 32 marks · Integration techniques
   $\displaystyle\int \frac{3x + 1}{x^2 - 1}\,dx =$
   • A.
     $2\ln|x - 1| + \ln|x + 1| + C$
   • B.
     $\ln|x - 1| + 2\ln|x + 1| + C$
   • C.
     $\tfrac{3}{2}\ln|x^2 - 1| + C$
   • D.
     $\tfrac{3}{2}\ln|x^2 - 1| + \arctan x + C$
4. Question 42 marks · Improper integrals
   $\displaystyle\int_2^{\infty} \frac{dx}{x(\ln x)^2}$ is
   • A.
     $\ln 2$
   • B.
     $\dfrac{1}{\ln 2}$
   • C.
     $\dfrac{1}{2}$
   • D.
     divergent
5. Question 52 marks · Differential equations
   Let $y = f(x)$ be the solution of $\dfrac{dy}{dx} = x + y$ with $f(0) = 1$. Using Euler's method with two steps of size $0.5$, the approximation to $f(1)$ is
   • A.
     $1.5$
   • B.
     $2$
   • C.
     $2.5$
   • D.
     $3$
6. Question 62 marks · Differential equations
   A population $P(t)$ satisfies the logistic differential equation $\dfrac{dP}{dt} = 3P - 0.006P^2$ with $P(0) = 50$. The limiting value of $P$ as $t \to \infty$ and the value of $P$ at which $P$ is growing fastest are, respectively,
   • A.
     500$ and $250

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 500$̲ and $250
   • B.
     500$ and $500

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 500$̲ and $500
   • C.
     1000$ and $500

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 5: 1000$̲ and $500
   • D.
     250$ and $125

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 250$̲ and $125
7. Question 72 marks · Parametric & vector motion
   A curve is given parametrically by $x(t) = t^2$ and $y(t) = \dfrac{2}{3}t^3$. The length of the curve from $t = 0$ to $t = \sqrt{3}$ is
   • A.
     $\dfrac{7}{3}$
   • B.
     $\dfrac{14}{3}$
   • C.
     $\dfrac{16}{3}$
   • D.
     $\dfrac{28}{3}$
8. Question 82 marks · Parametric & vector motion
   A particle moves in the $xy$-plane with velocity vector $\langle 2t,\ 3t^2 \rangle$ for $t \geq 0$. At time $t = 1$ the particle is at the point $(2, 5)$. Its position at time $t = 2$ is
   • A.
     $(3, 7)$
   • B.
     $(4, 8)$
   • C.
     $(5, 12)$
   • D.
     $(6, 13)$
9. Question 92 marks · Series convergence
   The interval of convergence of the power series $\displaystyle\sum_{n=1}^{\infty} \frac{(x - 2)^n}{n \cdot 3^n}$ is
   • A.
     $(-1, 5)$
   • B.
     $[-1, 5)$
   • C.
     $(-1, 5]$
   • D.
     $[-1, 5]$
10. Question 102 marks · Series error bounds
    The series $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} = 1 - \frac{1}{4} + \frac{1}{9} - \frac{1}{16} + \cdots$ converges to a sum $S$. If the first four terms are used to approximate $S$, the alternating series error bound guarantees that the error is at most
    • A.
      $\dfrac{1}{16}$
    • B.
      $\dfrac{1}{25}$
    • C.
      $\dfrac{1}{36}$
    • D.
      $\dfrac{1}{5}$
11. Question 112 marks · Series error bounds
    A function $f$ has derivatives of all orders, and $\left|f^{(4)}(x)\right| \leq 6$ for all $x$ in $[0, 1]$. If the third-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f\!\left(\tfrac{1}{2}\right)$, the Lagrange error bound guarantees that the error is at most
    • A.
      $\dfrac{1}{16}$
    • B.
      $\dfrac{1}{32}$
    • C.
      $\dfrac{1}{64}$
    • D.
      $\dfrac{3}{8}$
12. Question 122 marks · Taylor series
    The third-degree Taylor polynomial for $f(x) = \ln x$ about $x = 1$ is
    • A.
      $(x - 1) - \dfrac{(x-1)^2}{2} + \dfrac{(x-1)^3}{3}$
    • B.
      $(x - 1) - \dfrac{(x-1)^2}{2!} + \dfrac{(x-1)^3}{3!}$
    • C.
      $(x - 1) + \dfrac{(x-1)^2}{2} + \dfrac{(x-1)^3}{3}$
    • D.
      $1 + (x - 1) - \dfrac{(x-1)^2}{2} + \dfrac{(x-1)^3}{3}$
13. Question 139 marks · Taylor series
    Series. Let $f$ be the function defined by $f(x) = e^{-x^2}$, and let $g$ be the function defined by $g(x) = \displaystyle\int_0^x f(t)\,dt$. You may use the fact that the Maclaurin series for $e^u$ is $\displaystyle\sum_{n=0}^{\infty} \frac{u^n}{n!}$, which converges for all real $u$.
    (a)

    Write the first four nonzero terms and the general term of the Maclaurin series for $f$. Use the series to find $f^{(6)}(0)$.

    [3]
    (b)

    Write the first four nonzero terms and the general term of the Maclaurin series for $g$.

    [2]
    (c)

    Use the first two nonzero terms of the series in part (b) to approximate $g\!\left(\tfrac{1}{2}\right)$. Show that this approximation differs from the true value of $g\!\left(\tfrac{1}{2}\right)$ by less than $\dfrac{1}{300}$.

    [2]
    (d)

    Find $\displaystyle\lim_{x \to 0} \frac{g(x) - x}{x^3}$.

    [2]
14. Question 149 marks · Polar curves
    Polar curves. For $0 \leq \theta \leq \pi$ the polar curve $r = 4\sin\theta$ is the circle of radius $2$ centered at the point $(0, 2)$, and the polar curve $r = 2$ is the circle of radius $2$ centered at the origin. The two circles cross at two points above the $x$-axis. Let $R$ be the region that lies inside the circle $r = 4\sin\theta$ and outside the circle $r = 2$.
    (a)

    Find all values of $\theta$ in $[0, \pi]$ at which the two curves intersect.

    [2]
    (b)

    Find the area of $R$.

    [3]
    (c)

    Find the slope of the line tangent to the curve $r = 4\sin\theta$ at the point where $\theta = \dfrac{\pi}{6}$.

    [2]
    (d)

    A particle moves along the curve $r = 4\sin\theta$ so that $\dfrac{d\theta}{dt} = 3$ for all $t$. Find the rate of change of the particle's distance from the origin with respect to time at the instant when $\theta = \dfrac{\pi}{3}$.

    [2]

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