Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \( R(x) \), and cost, \( C(x) \), are in thousands of dollars, and \( x \) is in thousands of units. \( R(x)=8 x-4 x^{2}, C(x)=x^{3}-5 x^{2}+3 x+1 \)

Find the integral. \( \int\left(x^{4}-2 x^{3}\right)^{4}\left(4 x^{3}-6 x^{2}\right) d x \)

\( 1 \leftarrow \) Suppose the average daily metabolic rate for certain types of animals can be expressed as a function of mass by \( \mathrm{r}=132.8 \mathrm{~m} \).3.3 , where m is the mass of the animal (in kilograms) and r . a. If the mass of an animal is changing at a rate of \( \frac{\mathrm{dm}}{\mathrm{dt}} \), find \( \frac{\mathrm{dr}}{\mathrm{dt}} \). b. Determine \( \frac{\mathrm{dr}}{\mathrm{dt}} \) for a \( 120-\mathrm{kg} \) animal that is gaining mass at a rate of 3 kg per day. \( \frac{\mathrm{dr}}{\mathrm{dt}}= \)

Approximate the area under the graph of \( f(x) \) and above the \( x \)-axis using \( n \) rectangles. \( f(x)=x^{2}+2 \); interval \( [0,5] ; n=5 \); use left endpoints A. 40 B. 32 C. 65 D. 66

Graph the polynomial function \( f(x)=(x+1)^{2}(x-6)^{2} \) using parts (a) through (e). The graph of \( f \) behaves like \( y=x^{4} \) for large values of \( |x| \). (b) Find the \( x \) - and \( y \)-intercepts of the graph of the function. The \( x \)-intercept(s) is/are \( -1,6 \). (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) The \( y \)-intercept is 36 . (Simplify your answer. Type an integer or a fraction.) (c) Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the \( x \)-axis at each \( x \)-intercept. The zero(s) of \( f \) is/are (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)

91. \( f(\theta)=\frac{6\left(2-\theta^{3}\right)}{3-2 \theta} \)