The speed of a car traveling in a straight line is reduced from 45 to 30 miles per hour at a distance of 264 feet. Find the distance in which the car can be brought to rest from 30 miles per hour, assuming the same constant deceleration.

In this exercise, use the properties of summation and Theorem $5.2$ to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result.

$$\displaystyle\sum_{i=1}^{15}(2 i-3)$$

In this exercise, find the indefinite integral and check the result by differentiation.

$$\int(5-x) d x$$

In this exercise, evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result.

$$\int_{-3}^3 v^{1 / 3} d v$$

In this exercise, find the indefinite integral and check the result by differentiation.

$$\int t^3 \sqrt{t^4+5} d t$$

In this exercise, find the integral.

$$\int \frac{3}{2 \sqrt{x}(1+x)} d x$$

In this exercise, approximate the definite integral using the Trapezoidal Rule and Simpson's Rule with $n=4$. Compare these results with the approximation of the integral using a graphing utility.

$$\int_0^{\sqrt{\pi / 4}} \tan x^2 d x$$

In this exercise, find the derivative of the function.

$$y=\operatorname{coth} 3 x$$

In this exercise, find the indefinite integral.

$$\int \frac{x^3-6 x-20}{x+5}\ dx$$

Show that $\int_0^1 x^a(1-x)^b d x=\int_0^1 x^b(1-x)^a d x$.

The profit P for a company is $P=100 x e^{-x / 400}$, where x is the number of units sold. Approximate the change and percent change in profit as sales increase from x=115 to x=120 units.

Show that $\int_0^1 x^2(1-x)^5 d x=\int_0^1 x^5(1-x)^2 d x$.

Consider the functions $f$ and $g$, where $f(x)=6 \sin x \cos ^2 x \quad$ and $\quad g(t)=\int_0^t f(x) d x$.

Consider the function

$$h(t)=\int_{\pi / 2}^t f(x) d x .$$

Use a graphing utility to graph $h$. What is the relationship between $g$ and $h$ ? Verify your conjecture.

Consider the functions $f$ and $g$, where $f(x)=6 \sin x \cos ^2 x \quad$ and $\quad g(t)=\int_0^t f(x) d x$.

Does each of the zeros of $f$ correspond to an extremum of $g$ ? Explain.