AP Calculus AB (Section 1, Part B)

Consider the graph of a function f as shown below:

F\prime\left(x\right)=\frac{x^3+\cos{x}}{x^2-3}dx

If F(x) is the antiderivative of the function F\prime(x) , then what is the value of F(7) – F(5)?

Consider a continuous function g that is positive and has domain x > 0. If the asymptotes of the graph of g are x = 0 and y = 3, then which of the following is true?

A car’s speed is modeled by the differentiable function g(t), where t is the time in hours since the start of the journey and g(t) is measured in kilometers per hour. Which of the following is the best interpretation of g\prime(3) = 15.2?

The function f has the first derivative given by f\left(x\right)=-x^5+x^4+2x^3-2x^2-x+1 . On what intervals is the graph of f concave up?

Consider the graph of functions f, g, and h above. Which of the following statements is true?

A car moves along a straight road with velocity v(t)=4t^2\cos{(t)} kilometer per hour. Determine the total distance, in kilometers, covered by the car during the time interval 0\le t\le3 hours.

Consider the derivative of a function f:

f^\prime\left(x\right)=\frac{2(x^2-1)}{(x^2+x+1)^2}

At what value of x does the given function achieve a relative minimum?

The population of a city grows at the rate of P\prime(t) people per year. If there are 500,000 people in the city at time t = 0, which of the following expressions gives the population of the city at time t = 5 years?

Consider a function h that is continuous on the closed interval [0,3] with h(0) = 2 and h(3) = 6. Of the following conditions, which would guarantee that there is a number d in the open interval (0, 3) where h\prime(d)=2 ?

p"(x)=x(x+3)^2(x-5)^3
q"(x)=x(x+3)^2(x-5)^3+2
r"(x)=x(x+3)^2(x-5)^3-2

Consider the double differentiation of the functions p, q, and r as shown above. Which of the functions p, q, and r have a graph with exactly two points of inflection?

Consider a different set of values for a function g(x) at selected values of x. If g(x) is twice-differentiable on the interval 1\le x\le5 , which of the following statements could be true?

Consider a defined function p\left(x\right)=\ln{(2+x^2)} , and another defined function q\left(x\right)=x^7+x^4 . The line tangent to the graph of p at x = 2 is parallel to the line tangent to the graph of g at x = b, where b is a positive constant. What is the value of b?

Consider a continuous function g for all real numbers. Also, consider another function h which is defined as h\left(x\right)=\int_{2}^{x}{g(t)dt} . If the average rate of change of the function h on [3, 6] is 10, which of the following statements is true?

Consider the function f(x) defined on the interval [a, b]. If f(x) is continuous on [a, b] and differentiable on (a, b), which of the following statements is true?