AP Calculus AB Practice Set 1

Consider two functions f(x) and g(x), and \left[f\prime\left(4\right)\right]^2-4\left[g^\prime\left(4\right)\right]^2=0 .

Evaluate \lim_{x\rightarrow4}{\frac{f\left(x\right)-f\left(4\right)}{g\left(x\right)-g\left(4\right)}} .

Find the particular solution of the differential equation log\left(\frac{dy}{dx}\right)=5x+6y at x = 0,y = 0.

Calculate the volume of the solid of revolution enclosed by the graphs of f\left(x\right)=x^2-6x+3,x=0 , and x = 3 and rotated around the x-axis using slicing method.

What is the average value of g\left(x\right)=6x^2-2x over the interval \left[-2,3\right] ?

Is the function given below continuous at x = 2? Why or why not?

f\left(x\right)=7x-2

Compute the limit and find the horizontal asymptote of the expression:

\lim_{x\rightarrow\infty}{f\left(x\right)=}\lim_{x\rightarrow\infty}\frac{2x^2+3x-5}{4x^3+x^2}

Let f\left(x\right)=csin\left|x\right|+de^{|x|} is differentiable at x = 0 if

The function f has a vertical asymptote at x = 8, a horizontal asymptote at y = -11, and f(0) = 0.

What is the possible rational function that meets the given conditions?

A trough, 12 feet in length, has isosceles triangle-shaped ends with a top width of 4 feet and a height of 1 foot. Water is being poured into the trough at a rate of 9 cubic feet per minute. Determine the rate at which the water level is rising when the water depth is 7 inches.

Given a continuous function f, the right Riemann sum estimate for the integral \int_{0}^{4}f\left(x\right)dx with n subintervals of equal length is expressed as \frac{4\left(n+2\right)\left(5n+3\right)}{n^2} for all positive integers n. Determine the exact value of the integral \int_{0}^{4}f\left(x\right)dx .