Directions: Do the following problems and follow the instructions. Add space to insert your work as desired, but clearly fill in the blanks and outline your answers where appropriate. Illegible writing won’t be given partial credit and show all work. No work no credit. Five (5) points will be deducted if your name is not typed or written at the top of each page and/or you don’t write your answers on this answer sheet. .

1. Verify that F(x) is an antiderivative of the integrand f(x) and use Part 2 of the Fundamental Theorem to evaluate the definite integrals.

Show work.

2. Use Leibniz’s Rule to solve the following:

3. Find the area of the shaded region (yellow) in Figure 1.

Figure 1

4. Represent the volume in figure 2 as an integral and evaluate the integral.

Figure 2

5. Calculate the length of

6. The conical container in Figure 3 is filled with loose grain which weighs 40 pounds/ft3.

a. How much work is done lifting all of the grain over the top of the cone?

b. lifting the top 4 feet of grain over the top of the cone?

Figure 3

7. A function is one-to-one if each horizontal line intersects thegraph of a function:

a. Once

b. Two time

c. Three times

d. It asymptotically approaches the horizontal line.

8. Given two sets of real numbers, domain (starting set) and range (target set), a function is:

a. A rule for which each element of the domain is assigned to one or more elements in the range.

b. A rule that assigns two or more elements of the domain to one and only one element of the range.

c. A rule for which each element of the domain corresponds to one and only one element in the range.

d. None of the above

9. Which of the following tests is a function guaranteed to pass? Select all that apply.

ð The Kirlian line test

ð Horizontal line test

ð Spectral line test

ð Vertical line test

10. In Fig 4, find the following exactly.

· ? = ________

· Sin(q) = ________

· Cos (q) = ________

· Tan(q) = ________

· Csc(q) = ________

· Sec(q) = ________

· Cot(q) = ________

11. In Fig. 4, angle θ is

· the arcsine of what number? = ________

· the arctangent of what number? = ________

· the arcsecant of what number? = ________

· the arccosine of what number? = ________

12.

Rocket Launch. You are observing a rocket launch from a point 8000 feet from the launch pad (Fig. 5). When the observation angle is π/6, the angle is increasing at π/6 feet per second. How fast is the rocket traveling? (Hint: θ and h are functions of t.)

13. Evaluate the integrals

14. Decompose the fraction

Logistic Growth: The growth rate of many different populations depends not only on the number of individuals (leading to exponential growth) but also on a “carrying capacity” of the environment. If x is the population at time t and the growth rate of x is proportional to the product of the population and the carrying capacity M minus the population, then the growth rate is described by the differential equation where k and M are constants for a given species in a given environment.

15. Let k = 1 and M = 100, and assume the initial population is x(0) =

5 .

a. Solve the differential equation dx/dt = x(100 – x) with x(0)=5 .

b. Graph the population x(t) for 0 ≤ t ≤ 20.

c. When will the population be 20? 50? 90? 100?

d. What is the population after a “long” time? (Find the limit, as t becomes arbitrarily large, of x)

16. Write 0.99999…. using sigma notation.

17. The P–Test to determine whether the given

series converges, and then (b) use the Integral Test to verify your convergence

conclusion of part (a).

18. Determine whether the given series converge or diverge.

19. Determine whether the given series Converge Absolutely, Converge Conditionally, or Diverge and give reasons for your conclusions.

20. Evaluate the following integral to answer the riddle: What does a forester put on his toast?

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