review for Test 3

UPDATED FOR ERRORS on 12-6

Test 3 will cover sections 4.7 - 5.7

Optimization

• Find two nonnegative numbers whose sum is $16$ and so that the product of one number and the square of the other number is a maximum.

• An oil rig is 2 miles offshore, the refinery is 3 miles down shore. A pipe is run connecting the rig to the shore and then run along the shore line. The cost per mile in the water is 5M and the cost per mile on the shore is 2M. Find the point along the shore where the total cost is minimal.

As an example, if the pipe goes directly from the rig to the refinery the total cost would be:

$$~ \sqrt{2^2 + 3^2}\cdot 5 + 0 \cdot 2 ~$$

Whereas, if the pipe went directly to shore, then 3 miles along the shore the cost would be

$$~ 2 \cdot 5 + 3\cdot 2 ~$$

• Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 20 feet, what dimensions will result in the rectangle having largest possible area?

Approximating area

• Let $f(x) = x^3$. We wish to estimate the area under $f(x)$ between $1$ and $3$. We do so with $L_4$. That is, we split the interval into 4 equal sized pieces through points $x_0, x_1, x_2, x_3, x_4$ and then for each piece, we use a rectangle based on the left-hand endpoint. What is the estimate? Write a formula for $L_n$ for any $n$. If we were to take $n \rightarrow \infty$ what would be the limit of $L_n$?

• Using the grid, give a lower bound and an upper bound to the area under the curve.

• Based on the following graph, find $\int_0^4 f(x) dx$, $\int_0^1 f(x) dx$, $\int_1^3 f(x) dx$, $\int_4^3 f(x) dx$.

• Based on the following graph, find $\int_0^2 f(x) dx$, $\int_0^6 f(x) dx$.

• We know $\int_0^3 f(x) dx = 3$, $\int_2^4 f(x) dx =3$ and $\int_3^4 f(x) dx = 3$. Find $\int_0^2 f(x) dx$ and $\int_2^3 f(x)dx$.

• We know $\int_a^b f(x) dx = 5$ and $\int_b^a g(x) dx = 6$. Find $\int_a^b (7f(x) + 8g(x)) dx$.

• Find

$$~ \int (x^3 + 3x^2 + 2x + 1/x) dx, \quad \int (\sin(x) + \cos(x)) dx, \quad \int (e^x + e^{-x}) dx $$~ * Find the indefinite integrals $$~ \int 4(x+5)^6 dx, \quad \int x \sqrt{1 + x^2} dx, \quad \int \frac{\ln(x)}{x} dx ~$$

• Find the definite integrals

$$~ \int_1^3 (-15x^2 + 12x) dx, \quad \int_{\pi/6}^{\pi/3} \sec^2(x) dx, \quad \int_{-1}^1 10e^x dx ~$$

• Find the definite integrals

$$~ \int_0^1 \frac{x}{10}e^{-10x^2} dx, \quad \int_0^1 5x \sqrt{1 - x^2} dx, \quad \int_0^\pi e^{\sin(x)} \cos(x) dx ~$$

• If $1/(2-x) < x^x < x^2 - x + 1$ for $0 < x < 1$, what are upper and lower bounds for $\int_{1/2}^1 x^x dx$?

• The function $A(x) = \int_0^x f(u) du$, where $f(u)$ is given the graph below. Answer the following:

On what intervals is $A(x)$ increasing? What are the critical values of $A(x)$? Does the first derivative test say that $x=2$ is a relative maximum?

• The function $Ein(x)$ is defined by $\int_0^x (1 - e^{-t}) dt/t$. (Assuming the function is defined by continuity at $t=0$.) Given this find values for $Ein'(3)$ and $[Ein(x^2)]'$.