Determine \( \int y d x \) if: \( 2.1 \quad y=\left(x^{2}-2 x+1\right) e^{2 x} \) 2.2 2.3 \( y=\sin ^{3} 4 x \cos ^{2} 4 x \) 2.4 \( y=\cot ^{5} 3 x \) \( 2.5 \quad y=\cos ^{2}\left(\frac{5 x}{4}\right) \)

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 \( f(x)=2 x^{4}+12 x^{3}-18 x^{2}-108 x \). Answer parts (a) through (e). (a) Determine the end behavior of the graph of the function. The graph of \( f \) behaves like \( y=\square \) for large values of \( |x| \). (b) Find the \( x \) - and \( y \)-intercepts of the graph of the function. The \( x \)-intercept(s) is/are \( \square \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) The \( y \)-intercept is \( \square \). (Type an integer or a simplified 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 \( ~ \) is/are (Tvne an intener \( n \).

Explique o Teorema Fundamental do Cálculo e como ele relaciona a diferenciação e a integração.

\( 1 \leftarrow \) 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. \( \frac{\mathrm{dr}}{\mathrm{dt}}= \)

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