AP Calculus AB (Section 1, Part A)

Find y’ given that y=3\left(x-4\right)^2-2 .

Evaluate the Integral I = \int{\frac{18}{t\left(t^6-3\right)}dt} .

Find the derivative of the function f(t).

f(t) = \frac{t^4+1}{\left(t^4-1\right)^6}

Use the Trapezoidal Rule to approximate the area under the curve of the differentiable function f\left(x\right) on the interval \left[1,\ 5\right] with n=4,\ as given in the table below:

If f\left(x\right)=cos\left(x^2-\pi\right) , then f\prime\left(\sqrt{\frac{\pi}{2}}\right) .

Find the average rate of change of the function f\left(x\right)=x^2+5x from x_1=3 to x_2=8 .

If \int_{0}^{6}f\left(x\right)dx=10,\ \int_{3}^{0}g\left(x\right)dx=-5 , and \int_{3}^{6}f\left(x\right)dx=6 . Find the value of \int_{0}^{3}\left(2f\left(x\right)+5g\left(x\right)\right)dx .

Determine the equation of the tangent line to the function x^7+x^5y^4=0 at point (-1, -1) in the form y = mx + b.

Determine the area enclosed by the curves represented by the provided equations.

y=2x^2-4x+4;\ y=x^2+6x-5

Evaluate the definite integral: \int_{-\infty}^{0}{\frac{z}{z^4+16}dz} .

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.

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 .