AP Calculus AB (Section 2, Part B)

Consider the function g below:

g\left(x\right)=\frac{e^{2x}}{\sqrt{1+x^2}}

(a) Find the domain of g.
(b) Determine whether g is an even function, an odd function, or neither. Justify your conclusion.
(c) At what values of x does g have a relative maximum or a relative minimum? For each such x, use the first derivative test to determine whether g(x) is a relative maximum or a relative minimum.
(d) Find the range of g.

Suppose g(x) is a differentiable function defined on the closed interval [0, 5] with the following information:

The function g satisfies g(0) = 15, and the first derivative g\prime(x) satisfies the inequality 0\le g\prime(x)\le5 for all x in the closed interval [0, 5]. The function g has a continuous second derivative for all real numbers.

(a) Use a midpoint Riemann sum with three subintervals of equal length indicated by the data in the table to approximate the value of g(5).
(b) Determine whether the actual value of g(5) could be 50. Explain your reasoning.
(c) Evaluate \int_{2}^{4}{g"(x)dx} .
(d) Find \lim_{x\rightarrow0}{\frac{g\left(x\right)-15e^x}{0.5g\left(x\right)-7.5}} .

Consider the curve given by the equation:

4x^2+3y^3=5xy

(a) Show that \frac{dy}{dx}=\frac{5y-8x}{9y^2-5x} .
(b) Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.
(c) Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.
(d) A particle is moving along the curve. At the instant when the particle is at the point (-1, 2), its horizontal position is increasing at a rate of \frac{dx}{dt}=4 units per second. What is the value of \frac{dy}{dt} , the rate of change of the particle’s vertical position, at that instant?

Consider a scenario where a water tank is being filled at a varying rate. The volume of water in the tank is modeled by the function V(t) = 16t^2-4t^3 for 0\le t\le4 , where V(t) is measured in cubic meters and t is measured in seconds.

(a) At the instant that the water tank has a volume of 8 cubic meters, the volume is increasing at a rate of \frac{1}{2} cubic meters per second. Find the rate at which the water level is changing at that instant. Indicate units of measure.
(b) The volume of the water tank is modeled by V(t) = 16t^2-4t^3 for 0\le t\le4 . At what time t is the rate of change of the water volume equal to 2 cubic meters per second?
(c) A model for the volume of another water tank is given by the function W for 0\le t\le4 , where W(t) is measured in cubic meters and t is measured in seconds. The rate of change of the volume of the second water tank is given by W^\prime\left(t\right)=2t^2+3t+1 . Based on this model, by how many cubic meters does the volume of the second water tank increase from time t = 0 to t = 2?