Suppose that the trapezoidal rule is used to estimate the value of $\int_{0}^{0.2} \int_{-0.1}^{0.1} e^{x^{2}+2 y} d y d x$. Determine the smallest value of n so that the resulting approximation $T_{n, n}$ is accurate to within $10^{-4}$ of the actual value of the integral.

In this exercise,

a. List all possible rational zeros.

b. Use synthetic division to test the possible rational zeros and find an actual zero.

c. Use the quotient from part (b) to find the remaining zeros of the polynomial function.

$$f(x)=x^3-4 x^2+8 x-5$$

Use Descartes' rule of signs to determine the possible number of positive and negative real zeros for each function. Then, use a graph to determine the actual numbers of positive and negative real zeros.

$$P(x)=3 x^4+2 x^3-8 x^2-10 x-1$$

Use the Rational Zero Test to list all possible rational zeros of f. Then use a graphing utility to graph the function. Use the graph to help determine which of the possible rational zeros are actual zeros of the function. $f(x)=-4 x^{3}+8 x^{2}-3 x+15$

Calculate the percentage of the points scored for each region based on your results in previous part and compare them to the probabilities you just calculated recently. How closely did the results from spinning match the actual probabilities? Explain any large differences.

Use the Midpoint Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. $\int_{0}^{\pi} x^{2} \sin x d x, \quad n=8$

You're visiting the shark tank at the aquarium when you see a 2.5-m-long shark that appears to be swimming straight toward you at $2.0 \mathrm{~m} / \mathrm{s}$. What is the shark's actual speed through the water? You can ignore the glass wall of the tank.

In this exercise,

a. List all possible rational zeros.

b. Use synthetic division to test the possible rational zeros and find an actual zero.

c. Use the quotient from part (b) to find the remaining zeros of the polynomial function.

$$f(x)=2 x^3+6 x^2+5 x+2$$

Use the Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. $\int_{0}^{1} e^{-\sqrt{x}} d x, \quad n=6$

(a) What mass of ${}_{92}^{235}\mathrm{U}$ was actually fissioned in the first atomic bomb, whose energy was the equivalent of about $20$ kilotons of TNT $(1$ kiloton of TNT releases $5\times10^{12} \mathrm{~J})?$ (b) What was the actual mass transformed to energy?

Use the Midpoint Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. $\int_{0}^{\pi} x^{2} \sin x d x,\ n=8$

A blueprint for a museum uses a scale of $\frac{1}{4}$ in. 1 ft. One of the rooms on the blueprint is $3 \frac{3}{4}$ in. long. How long is the actual room? F. 4 ft G. 15 ft H. 45 ft J. 180 ft

In this exercise

a. List all possible rational roots.

b. Use synthetic division to test the possible rational roots and find an actual root.

c. Use the quotient from part (b) to find the remaining roots and solve the equation.

$$x^3-2 x^2-11 x+12=0$$

A MOSFET has a rated power of $50 \mathrm{~W}$ and a maximum specified junction temperature of $150^{\circ} \mathrm{C}$. The ambient is $T_{\text {amb }}=25^{\circ} \mathrm{C}$. Find the relationship between the actual operating power and $\theta_{\text {case-amb. }}$.