AP Calculus BC Full-length practice exam 1

Using the Left Riemann Sum with $n=4$ equal subintervals, approximate ${\int }_0^2{x}^{2}dx$.

For $g(x)=x^4-8x^2$, use the Second Derivative Test to classify $x=0$ and $x=\pm 2$.

The Squeeze Theorem guarantees ${\lim }_{x\to 0}{x}^2\sin \left(\frac{1}{x}\right)=0$ because:

Find the interval of convergence of the power series $\sum _{n=0}^{\infty }\frac{x^n}{n+1}$.

Solve the separable ODE $\frac{dy}{dx}=3x^{2}y$ with initial condition $y(0)=2$.

Does the geometric series $\sum _{n=0}^{\infty }{\left(\frac{2}{3}\right)}^n$ converge? If so, find its sum.

Let $f$ be twice differentiable with $f^{\prime }(2)=0$, $f^{\prime \prime }(2)=0$, and $f^{\prime \prime \prime }(2)\ne 0$. What can be concluded about $x=2$?

The derivative of $f(x)=\frac{x^2+1}{x-3}$ at $x=4$ is:

Find $\frac{d}{dx}\left[\arctan \left(\frac{y}{x}\right)\right]$ treating $y$ as a differentiable function of $x$.

For what value of $c$ is $f(x)=\{\begin{array}{ll}{x}^2+c& x<2\\ 3x-1& x\ge 2\end{array}$ continuous at $x=2$?

What is $\frac{d}{dx}[\ln (x^3)]$ for $x>0$?

A curve is given parametrically by $x(t)=t^2-1$ and $y(t)=2t$. Find $\frac{dy}{dx}$ in terms of $t$.

Find $\frac{d}{dx}\left[\tan x\right]$ directly from $\sin x/\cos x$ using the quotient rule.

Find the volume of the solid formed by revolving $y=\sqrt{x}$ on $[0,4]$ around the $x$-axis using the disk method.

Use L'Hôpital's Rule to evaluate ${\lim }_{x\to 1}\frac{x^3-3x+2}{x^2-2x+1}$.

For $f(x)=x^3-6x^2+9x$, on what intervals is $f$ increasing?

If $f(x)=e^{2x}\cos x$, find $f^{\prime }(x)$.

A particle has acceleration $\mathbf{a}(t)=⟨2,6t⟩$ with $\mathbf{v}(0)=⟨1,0⟩$ and $\mathbf{r}(0)=⟨0,0⟩$. Find $\mathbf{r}(1)$.

Find the volume of the solid generated by revolving $y=e^x$ on $[0,1]$ around the $x$-axis.

Find the linearization $L(x)$ of $f(x)=\sqrt{x}$ at $a=9$, and use it to approximate $\sqrt{9.4}$.

If $F(x)={\int }_0^x\cos (t^2)dt$, find $F^{\prime }(x)$ by the Fundamental Theorem of Calculus Part 1.

Evaluate $\int \sin x\cos xdx$.

A farmer has $200$ m of fencing to enclose a rectangular field against a straight river (no fence on the river side). What dimensions maximize the enclosed area?

Find $\frac{d}{dx}\left[\sin (x^3)\right]$.

Find $\frac{d}{dx}\left[x^3\ln x\right]$ for $x>0$.

Let $f$ be continuous on $\mathbb{R}$ with $f(x)=\frac{\sin (x^2)}{x}$ for $x\ne 0$. What must $f(0)$ equal?

Solve the separable ODE $\frac{dy}{dx}=\frac{x}{y}$ with $y(0)=3$.

Find the slope of the tangent line to the polar curve $r=1+\sin \theta$ at $\theta =\pi /2$.

Does the $p$-series $\sum _{n=1}^{\infty }\frac{1}{n^{3/2}}$ converge or diverge?

Find the Maclaurin series for $f(x)=e^{-x}$ up to the $x^3$ term.

The region bounded by $x=y^2$ and $x=4$ is revolved around the $y$-axis. Find the volume.

Evaluate $\int (3x^2-4x+5)dx$.

Find the volume of the solid with a circular base of radius $2$ (centered at origin) and semicircular cross-sections perpendicular to the $x$-axis.

A particle's position is $s(t)=2\sin (\pi t)$ for $t\ge 0$. What is the particle's acceleration at $t=1$?

A particle moves with position $\mathbf{r}(t)=⟨t^2-1,2t⟩$. Find the speed at $t=2$.

If $f$ is differentiable at $x=a$, then ${\lim }_{x\to a}\frac{f(x)-f(a)}{x-a}$ equals $f^{\prime }(a)$. Using this definition, find $\frac{d}{dx}[x^{-1}]$ at $x=2$.

Using the limit definition, find $f^{\prime }(x)$ for $f(x)=3x^2-5$.

A point moves along the curve $y=x^2$. When $x=3$ the $x$-coordinate increases at $2$ units/sec. How fast is the distance from the origin increasing?

If $G(x)={\int }_1^{x^3}\ln tdt$, find $G^{\prime }(x)$.

Find the arc length of $y=\frac{2}{3}{x}^{3/2}$ from $x=0$ to $x=3$.

For the cardioid $r=1+\cos \theta$, find the total arc length.

On what interval is $f(x)=x{e}^{-x}$ concave down?

Let $f(x)=x^x$ for $x>0$. Find $f^{\prime }(x)$.

Find the Taylor series for $\ln (1+x)$ centered at $x=0$ (Maclaurin series).

The function $f(x)=\frac{x^2-4}{x-2}$ has what type of discontinuity at $x=2$?