Final Exam 1 covering Sections:



Final Exam 1: Sec.8.1&.3&.5, 7.1&.3, 9.2&.4

PCC Math 253, Calculus III Jeff Pettit, Instructor Final Exam Name:______________

Use of technology is encouraged to check your work or lead you to the answer, but please show a reasonable number of steps and please begin your work on this page. Show all work on additional page(s) if necessary. Put answers in the space provided.

8.1 Sequences

1. A fish farmer has 6,000 catfish in his pond. The number of catfish increases by 8% each month and the farmer harvests 200 catfish each month. The catfish population Pn after n months is given recursively as Pn = 1.08Pn-1 – 300 P0=6,000. How many catfish are in the pond after 5 months?

2. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? [pic]

8.3 Integral & Comparison Tests; Estimating Sums

3. Choose two below. Circle and correctly do the third for 1 point extra credit. Determine whether the series is convergent or divergent.

a. [pic]

b. [pic]

c. [pic]

4. Choose one below. Circle and correctly do the second for 1 point extra credit.

a. For the series [pic] find a value of n that will ensure that the error in the approximation s ( sn is less than 0.01

b. Use the sum of the first 5 terms of the following series to approximate the sum of the series, and estimate the error using the integral test: [pic]

8.5 Power Series

5. Choose two below. Circle and correctly do the third for 1 point extra credit. Find the radius of convergence and interval of convergence of the series.

a. [pic]

b. [pic]

c. [pic]

6. If [pic] is convergent, does it follow that [pic] is convergent? that [pic]is convergent? Justify your answer to both.

8.7 Taylor and Maclaurin Series

7. Choose two below. Circle and correctly do the third for 1 point extra credit. Find the Maclaurin series (by any method) and the radius of convergence.

a. f(x)=cos((x/2)

b. f(x)=x2ln(1+x3)

c. f(x)=xe-x

8. Expand [pic] as a power series and use your power series expansion to estimate [pic] correct to two decimal places.

9. Use multiplication or division of a power series to find the first three nonzero terms in the Maclaurin series for the following function: [pic]

7.1 Modeling with Differential Equations

10. Choose one below. Circle and correctly do the second for 1 point extra credit. Verify that each of the following equations is a solution to the given differential equation.

a. [pic] is a solution for [pic]

b. y = –t cos(t) – t is a solution for [pic] where y(() = 0

11. A population is modeled by the differential equation [pic]. For what values is the function increasing? Decreasing? What are the equilibrium solutions?

7.3 Separable Equations

12. Choose 2. Circle and correctly do the third for 1 point extra credit. Find the solution of the differential equation that satisfies the given initial condition.

a. [pic]

b. [pic]

c. [pic]

9.2 Vectors

13. Find the unit vector that has the same direction as 4i – 8j + k

9.4 The Cross Product

14. Choose two below. Circle and correctly do the third for 1 point extra credit. Find the cross product [pic] and verify that it is orthogonal to both a and b.

a. a = < 6, 0, –2 >, b = < 0, 8, 0 >

b. a = i – j – k, b = i + 2j + k

c. a = < t, t2, t3 >, b = < 1, 2t, 3t2 >

15. A wrench 30cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction < 0, 3, –4 > at the end of the wrench. Find the magnitude of the force needed to supply 100 N(m to the bolt.

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5% points each, 107% possible with extra credit

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Justify to the left.

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12.a

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14.a.

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