AP Calculus AB Full-length practice exam 2

The position of a particle is $s(t)=t^3-9t^2+24t$ for $t\ge 0$. At what time(s) is the particle at rest?

If a particle has velocity $v(t)=3t^2-6t$ (m/s), what is its displacement from $t=0$ to $t=4$?

Radioactive carbon-14 decays according to $\frac{dN}{dt}=-0.000121N$. If $N_0=1000$ atoms initially, how many atoms remain after $5730$ years (one half-life)?

The graph of $f^{\prime }$ is shown (not provided here — described): $f^{\prime }(x)>0$ for $x<2$, $f^{\prime }(2)=0$, and $f^{\prime }(x)<0$ for $x>2$. What can be concluded about $f$ at $x=2$?

If $f^{\prime }(x)>0$ on $(a,b)$, then $f$ is:

Find all inflection points of $f(x)=x^4-4x^3$.

Newton's law of cooling gives $\frac{dT}{dt}=-k(T-T_s)$, where $T_s=20$°C is room temperature. If an object cools from $100$°C to $60$°C in $10$ minutes, what is the temperature at $t=20$ minutes?

A function $f$ has $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)=0$. What can be concluded?

Use a left Riemann sum with 4 equal subintervals to approximate ${\int }_0^4{x}^{2}dx$.

Evaluate $\underset{x\to 3}{\lim }\frac{x^2-9}{x-3}$.

Find the area between $y=\sin x$ and $y=\cos x$ on $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$.

Differentiate $y={\tan }^{-1}\left(\frac{x}{3}\right)$ with respect to $x$.

Find $\frac{dy}{dx}$ if $y=(3x^2+1{)}^5$.

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

Differentiate $y=\ln (x^4+7)$ with respect to $x$.

Using the Second Derivative Test, classify the critical point $x=1$ of $f(x)=x^3-3x+2$.

If $y={\cos }^3(x^2)$, find $\frac{dy}{dx}$.

Use the limit definition of the derivative to find $f^{\prime }(2)$ for $f(x)=x^2+1$.

Evaluate $\int \frac{1}{x}dx$ for $x>0$.

Find the absolute maximum of $f(x)=-x^2+4x+1$ on the interval $[0,5]$.

Find $\int x\cos (x^2)dx$.

Find $f^{\prime }(x)$ if $f(x)=e^x\cos x$.

A slope field for $\frac{dy}{dx}=\frac{x}{y}$ is sketched. Which statement best describes a solution curve passing through $(0,3)$?

The volume of a cube is decreasing at $12$ cm³/s. How fast is the side length decreasing when the side is $2$ cm?

Find the volume of the solid with square cross-sections perpendicular to the $x$-axis, where the base is the region bounded by $y=\sqrt{x}$ and $y=0$ on $[0,4]$.

A particle moves so that $x(t)=t^3-6t^2+9t$ for $t\ge 0$. On which interval(s) is the particle's speed increasing?

A particle's velocity is $v(t)=t^2-4t+3$ for $0\le t\le 4$. What is the total distance traveled?

A population $P$ satisfies $\frac{dP}{dt}=0.05P$. If $P(0)=200$, find $P(t)$.

Use implicit differentiation to find $\frac{dy}{dx}$ for $x^2+y^2=25$.

A particle moves along a number line so that its position is $s(t)=t^3-6t+2$ (feet, $t$ in seconds). What is the velocity at $t=2$?

If $f$ is a differentiable function and $g(x)=[f(x){]}^3$, find $g^{\prime }(2)$ given that $f(2)=-2$ and $f^{\prime }(2)=5$.

If $f^{\prime \prime }(x)<0$ on an interval, then the graph of $f$ is:

A slope field for a differential equation shows short line segments with slope $x+y$ at each point $(x,y)$. Which differential equation corresponds to this slope field?

If $f(x)=\frac{x^2-4}{x-2}$ for $x\ne 2$ and $f(2)=5$, then $f$ is:

The general solution to $\frac{dy}{dx}=ky$ is $y=C{e}^{kx}$. If $y(0)=5$ and $y(2)=20$, find $k$.

Find $\frac{d}{dx}[\ln (\sin x)]$.

A bacteria culture doubles every 3 hours. If the initial count is $N_0$, what is the count after $t$ hours?

A farmer wants to enclose a rectangular plot next to a river (no fence needed along the river) using $200$ m of fencing. What dimensions maximize the enclosed area?

A tank contains $100$ gallons of water. Water drains at rate $r(t)=5t$ gal/min. How much water drains out in the first $4$ minutes?

A 5-foot tall person walks away from a 20-foot lamppost at $3$ ft/s. How fast is their shadow lengthening?

The function $f$ is continuous on $[2,8]$, differentiable on $(2,8)$, with $f(2)=1$ and $f(8)=13$. Which theorem guarantees a $c\in (2,8)$ where $f^{\prime }(c)=2$?

Let $f(x)=x^3-3x^2-9x+5$. On what interval(s) is $f$ increasing?

If $\underset{x\to 4^{-}}{\lim }f(x)=5$ and $\underset{x\to 4^{+}}{\lim }f(x)=7$, then:

Apply L'Hôpital's rule to evaluate $\underset{x\to 0}{\lim }\frac{e^x-1-x}{x^2}$.

A spherical balloon is being inflated so that its volume increases at $100$ cm³/s. How fast is the radius increasing when the radius is $5$ cm? (Recall $V=\frac{4}{3}\pi r^3$.)