A life insurance company needs a model to approximate the death rate of citizens during the years they are in the work force. The table gives the death rate $R$ per 1000 for individuals of age $x$. (Source: Department of Health and Human Services)

x	20	30	40	50	60
R	1.0	1.4	2.3	4.7	11.7

A model for these data is

$$\begin{aligned} & R=-91.1-6.313 x+0.035 x^2+45.794 \sqrt{x} \\ & 20 \leq x \leq 60 \end{aligned}$$

Use a graphing utility to plot the data and graph the model.

Find the average value of the function over the interval. Find the values of x where the function assumes its average value and graph the function.

Function: f(x)=x, Interval: [0,4]

Give an example of a function that is integrable on the interval [-1,1], but not continuous on [-1,1].

A company introduces a new product, and the profit in thousands of dollars over the first 6 months is approximated by the model

$$P=5(\sqrt{t}+30), \quad t=1,2,3,4,5,6 .$$

What, if any, is the advantage of using the approximation of the average given by the definite integral? (Note that the integral approximation utilizes all real values of t in the interval rather than just integers.)

A company introduces a new product, and the profit in thousands of dollars over the first 6 months is approximated by the model

$$P=5(\sqrt{t}+30), \quad t=1,2,3,4,5,6 .$$

Find the average value of the profit function by integration and compare the result with that in part (a). (Integrate over the interval [0.5,6.5].)

A company introduces a new product, and the profit in thousands of dollars over the first 6 months is approximated by the model

$$P=5(\sqrt{t}+30), \quad t=1,2,3,4,5,6 .$$

Use the model to complete the table and use the entries to calculate (arithmetically) the average profit over the first 6 months.

Determine whether the function

$$f(x)= \begin{cases}1, & x \text { is rational } \\ 0, & x \text { is irrational }\end{cases}$$

is integrable on the interval [0,1]. Explain.

Evaluate the definite integral. Use a graphing utility to verify your result.

$$\int_0^2 \frac{x}{\sqrt{1+2 x^2}} d x$$

Use a(t)=-32 feet per second per second as the acceleration due to gravity. (Neglect air resistance.)

Show that the height above the ground of an object thrown upward from a point $s_0$ feet above the ground with an initial velocity of $v_0$ feet per second is given by the function

$$f(t)=-16 t^2+v_0 t+s_0 .$$

Find the average value of the function over the interval. Find the values of x where the function assumes its average value and graph the function.

Function: $f(x)=x^3$, Interval: [0,2]

Write a program for a graphing utility to approximate areas by using the Midpoint Rule. Assume the function is positive over the indicated interval and the subintervals are of equal width. Use the program to approximate the area between the function and the x-axis over the indicated interval, and complete the table.

The graph of f is given in the figure. Determine the answers to parts (a) and (b) if the graph is translated two units upward.

The graph of f is given in the figure. Determine the average value of f on the interval [1,7].

The graph of f is given in the figure. Evaluate $\int_1^7 f(x) d x$.