AP Calculus AB Practice Set 2

Find the value of \frac{dy}{dx} by differentiating the equation x^4+\frac{1}{7}y^7=6-\frac{5}{3}x^3+y^4 .

The volume of air in a balloon, V(t), is mentioned by the function V(t) = 100 – 2t² for 0\le t\le5 , where t is the time in seconds. Find the rate, in cubic units per second, at which the volume of the balloon is changing at t = 4 seconds.

Solve the differential equation.

\frac{dy}{dx}=\frac{y}{x}-cot\frac{y}{x}cos\frac{y}{x} .

Under optimal laboratory conditions, a bacterial colony experiences exponential growth over time. After 2 hours, the population reaches 8,000 bacteria, and after 4 hours, it increases to 32,000. What was the initial number of bacteria in the colony?

The graph of function f is shown below:

Which of the following statements is false?

Consider a function f\left(x\right)=x\sqrt{4-x^2} . Find the interval on which the function is concave up.

If the function f\left(t\right)=-t^3+27t+37 represents the distance of a particle from the origin, at what distance the particle will come to rest.

Evaluate the integral by using the fundamental theorem of calculus \int_{x-2}^{x}{\frac{t+1}{t}dt} .

A continuous curve function y=f\left(x\right) is defined such that \lim_{x\rightarrow11}{\frac{f\left(x\right)+23}{x+7}=11} . Find the equation of the line to the curve that is normal to the tangent line at the point \left(x_1,y_1\right) .

If f(x) = \begin{cases} ax^2 – b & \text{if } x < 1 \\ -1x & \text{if } |x| \geq 1 \end{cases} is differentiable and continuous at x = 1. Find the values of a and b.