91Ó°ÊÓ

A particle moves with a velocity of \(v(t)\) \(\mathrm{m} / \mathrm{s}\) along an \(s\) -axis. Find the displacement and the distance traveled by the particle during the given time interval. (a) \(v(t)=2 t-4 ; 0 \leq t \leq 6\) (b) \(v(t)=|t-3| ; 0 \leq t \leq 5\)

Write each expression in sigma notation. but do not evaluate. $$1+\cos \frac{\pi}{7}+\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}$$

find the limits by making appropriate \- ubstitutions in the limits given in Theorem 7.9 .2 $$ \text { (a) } \lim _{x \rightarrow+\infty}\left(1+\frac{1}{3 x}\right)^{x} \text { (b) } \lim _{x \rightarrow 0}(1+x)^{1 / 3 x}$$

Use a calculating utility to find the left endpoint, right endpoint, and midpoint approximations to the area under the curve \(y=f(x)\) over the stated interval using \(n=10\) subintervals. $$y=\sqrt{x} ;[0,4]$$

Evaluate the definite integral two ways: first by a \(u\) -substitution in the definite integral and then by a \(u\) -substitution in the corresponding indefinite integral. $$\int_{0}^{\ln 5} e^{x}\left(3-4 e^{x}\right) d x$$

evaluate the integral, and check your answer by differentiating. $$\int x\left(1+x^{3}\right) d x$$

find \(g^{\prime}(x)\) using Part 2 of the Fundamental Theorem of Calculus, and check your answer by evaluating the integral and then differentiating. $$g(x)=\int_{1}^{x}\left(t^{2}-t\right) d t$$

(a) Express the sum of the even integers from 2 to 100 in sigma notation. (b) Express the sum of the odd integers from 1 to 99 in sigma notation.

A particle moves with a velocity of \(v(t)\) \(\mathrm{m} / \mathrm{s}\) along an \(s\) -axis. Find the displacement and the distance traveled by the particle during the given time interval. (a) \(v(t)=t^{3}-3 t^{2}+2 t ; 0 \leq t \leq 3\) (b) \(v(t)=e^{t}-2 ; 0 \leq t \leq 3\)

Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. $$\int_{4}^{9} 2 x \sqrt{x} d x$$