201-NYB-05 (Science) Final Exam, Winter 2012

1. (a) Find 11(4 x2) dx by taking the limit of a Riemann Sum. [4 marks]

(b) Check your answer using the Fundamental Theorem of Calculus (a.k.a. the Evaluation Theorem).[1 mark]

ni=1

c = cnni=1

i =n(n+ 1)

2

ni=1

i2 =n(n+ 1)(2n+ 1)

6

ni=1

i3 =[n(n+ 1)

2

]2

2. If 53

(f(x) 1) dx = 5 and 03

2f(x) dx = 3, find 50f(x) dx. [4 marks]

3. Evaluate each integral.

(a)

sin5 cot2 d [5 marks]

(b) 10

x arctanx dx [5 marks]

(c)

2x+ 1x3 + 6x2 + 9x

dx [5 marks]

(d)

x2

(9 x2)3/2 dx [5 marks]

(e)

1 +t

1t dt [5 marks]

4. Find the average value of the function g(x) = ex cos (1 ex) over the interval [ ln(1 + pi), 0].[5 marks]

5. Find the area of the region that is bounded by the graphs of y = 4x, y =1x

, and y = 1. [5 marks]

6. Let R be the region bounded by y = sinx and y = sinx from x = 0 to x = pi2 .(a) [3+3 marks] Set up the integral for the volume of the solid obtained by rotating R about the

y-axis:

i. by using the shell method [3 marks]ii. by using the disc/washer method [3 marks]

(b) Determine the volume by evaluating ONE of the integrals from part (a). [1 mark]

7. Determine the convergence of each improper integral. Determine the value of the integral if possible.

(a) pi/20

(1

x pi2 tanx

)dx [5 marks]

(b) 1

2x+ 5x2 + 1

dx [5 marks]

8. Find a general formula for the nth term of the following sequence (which begins with n = 1), assumingthe pattern continues: {

sin(pi7

)2

,sin(pi8

)6

,sin(pi9

)10

,sin(pi10

)14

, . . .

}.

Does this sequence converge or diverge? Justify your answer. [5 marks]

9. Determine if the seriesn=1

5en+1

23nconverges or diverges. If it converges, find its sum. [4 marks]

10. Determine if each series is absolutely convergent, conditionally convergent, or divergent. Clearly statewhich test you are using for each problem.

(a)n=1

lnnln(n2 + 1)

[5 marks]

(b)n=1

3 arctannn4 + 1

[5 marks]

(c)n=0

(1)n (2n)!5n (n!)2 [5 marks]

11. Find the interval of convergence of the power seriesn=1

(1)n(x 3)nn2

. [5 marks]

12. (a) Find the Taylor series for f(x) = lnx centered at x = 1.Note: Assume that f(x) has a power series expansion; do not show that Rn(x) 0. [4 marks]

(b) Use part (a) to show thatn=1

(1)n1n2n

= ln32

.

Note: You may assume that f(x) = lnx converges to its Taylor series if 0 < x < 2. [1 mark]

13. Let f(x) be a continuous function on the interval [a, b], where 0 < a < b, and let S be the regionbounded by f(x), x = a, x = b and the x-axis.

Suppose the volume of the solid formed by revolving S about the y-axis is 4pi and the volume of thesolid formed by revolving S about the line x = 1 is 8pi. Find the area of S. [5 marks]

ANSWERS

1.223

(x =2n, xi = 1 + 2in )

2.112

3. (a)15

cos5 13

cos3 + C

(b)pi

4 1

2

(c)19

xx+ 3 53(x+ 3) + C

(d)x

9 x2 arcsin(x

3

)+ C

(e) (1t)2 + 6(1t) 4 ln |1t|+ C4. 0

5. ln 2 38

6. (a) i. pi/20

4pix sinx dx

ii. 2 10

pi

(pi2

4 (arcsin y)2

)dy

(b) 4pi

7. (a) , diverges(b) , diverges

8. an =sin( pin+6 )4n 2 ; converges to 0

9. converges to5e2

8 e10. (a) diverges (Test for Divergence or nth term test)

(b) absolutely converges (Limit Comparison Test with

1n2 )

(c) absolutely converges (Ratio Test, L = 45 )

11. [2, 4]

12. (a)n=1

(1)n+1n

(x 1)n

(b) Plug x = 32 into lnx and its Taylor series

13. 2