calculus 2

Houston Community College

Department of Mathematics

Calculus I [Math 2414] Final Examination Review

You must show logical steps to support your choice Solve the problem.

1) Apply laws of logarithms to simplify the function f(x) = ln 9 - x2

4 + x2 . Then find its

derivative.

1)

Evaluate the integral.

2) 1/3

0

5x dx

9 - x4 ∫

A) π

8 B)

5

2 sin-1

1

27 C)

π

2 D) π

2)

3) dx

(x + 2) x2 + 4x + 3 ∫ 3)

4) dx

-x2 - 10x - 16 ∫ 4)

5) (sin-1 x)

5

1 - x2 ∫ dx 5)

Find the derivative of y with respect to x.

6) y = -sin-1 (11x2 + 4) 6)

7) y = tan-1 (ln 5x) 7)

Evaluate the integral.

8) 5 dx

36 - 25x2 ∫ 8)

Solve the problem.

9) Given the function y = x2x, find dy/dx by logarithmic differentiation. 9)

Find the area of the shaded region.

10)

x -π

-π

2

π

2 π

y

2

1

-1

-2

x -π

-π

2

π

2 π

y

2

1

-1

-2

y = cos2 x

y = -cos x

10)

Find the volume of the solid generated by revolving the shaded region about the given axis.

11) About the x-axis

x1 2 3 4

y20

18

16

14

12

10

8

6

4

2

x1 2 3 4

y20

18

16

14

12

10

8

6

4

2

y = 16 - x2

11)

Find the length of the curve.

12) y = 3x3/2 from x = 0 to x = 5

9 12)

Find the Cartesian coordinates of the given point.

13) (-4, -π/3) 13)

14) Evaluate the indefinite integral: x2∫ ex dx A) 2xex + 2ex + C B) x2ex - 2xex - 2ex + C

C) x2ex - 2xex + ex + C D) x2ex - 2xex + 2ex + C

14)

2

Solve the problem.

15) Evaluate sin2∫ xcos2x dx.

A) sinx

8 -

sin4x

32 + C B)

x

8 -

sin4x

32 + C

15)

16) Evaluate: 1

4 + x2 ∫ dx

A) 1

2 tan

-1 ( x

2 ) + C B) ln 4 + x2 | + C

C) ln | 4 + x2 + x| + C D) 1 2

sec-1( x

2 ) + C

16)

Express the integrand as a sum of partial fractions and evaluate the integral.

17) 6x - 18

x2 - 4x - 5 dx∫ 17)

Evaluate the improper integral or state that it is divergent.

18) ∞

0

2dx

9 + x2 ∫ You must show all steps. 18)

Solve the problem.

19) Find lim

x→∞ 5x - 6

3x - 4 . Apply l'Ho

^ pital's rule as many times as necessary, verifying your results

after each application.

19)

20) Find lim

x→0

1 - sec x

x3 . Apply l'Ho

^ pital's rule as many times as necessary, verifying your

results after each application.

20)

21) Find lim

x→1

x + cos πx

x2 - 1 . Apply l'Ho

^ pital's rule as many times as necessary, verifying your

results after each application.

21)

22) Find lim x (tan x)

x→0 . 22)

3

23) Given cot-1x + cot-1y = π

4 , (a) find

dy

dx by implicit differentiation. Then find (b) the

equation of the line tangent to the graph at the point P(1, 2).

23)

24) Evaluate ex

1 + e2x ∫ dx. 24)

25) Evaluate 1

4 + 9x2 ∫ dx. 25)

26) Evaluate arcsin(2x)

1 - 4x2 ∫ dx. 26)

Find the limit of the sequence or determine that the limit does not exist.

27) an = 1 + 2

n

n You must show all steps. 27)

Find the sum of the series.

28) ∞

n=1

8 1

5

n ∑ You must show all steps. 28)

Use the integral test to determine whether the series converges.

29) ∞

n=1

6n

n2 + 4 ∑ You must show all steps. 29)

Use the ratio test to determine if the series converges or diverges.

30) ∞

n=1

8n

n! ∑ You must show all steps. 30)

Determine convergence or divergence of the alternating series.

31) ∞

n=1

(-1)n

n1/2 ∑ 31)

Find the interval of convergence of the series. Test endpoints

32) ∞

n=0

(x - 9)n

9 + 8n ∑ 32)

Solve the problem.

33) Write the Taylor polynomial with center zero and of degree 5 for the function

f(x) = sin x.

33)

4

34) Find Taylor's formula for the function f at a = 0. Find both the Taylor polynomial Pn(x) of

the indicated degree n and the remainder term Rn(x) .

f(x) = 1

x + 2 , n = 3

34)

35) Write the Taylor series with center zero for the function f(x) = 1n(1 + x2). 35)

Find an equation for the line tangent to the curve at the point defined by the given value of t.

36) x = sin t, y = 5 sin t, t = π

3 36)

Solve the problem.

37) Plot the point with the polar coordinates (-3, - 7π

6 ). Then find the rectangular coordinates. 37)

38) Express the rectangular equation xy = 2 in polar form. 38)

39) Express the polar equation r = 3sin 2θ in rectangular form. 39)

40) Eliminate the parameter in x = 4sin t and y = 7cos t. Then sketch the curve. 40)

5

Answer Key Testname: MATH 2414 HCCS FINAL EXAM REVIEW

1) 13x

x4 - 5x2 - 36

2) B

3) sec-1 (x + 2) + C

4) sin-1 x + 5

3 + C

5) (sin-1 x)

6

6 + C

6) -22x

1 - (11x2 + 4)2

7) 1

x(1 + (ln 5x)2)

8) sin-1 5

6 x + C

9) x2x(2 ln x + 2)

10) 2 + π

2

11) 7168

15 π

12) 335

243

13) (-2, 2 3)

14) D

15) B

16) C

17) 2ln x - 5 + 4ln x + 1 + C

18) π

3

19) 5

3

20) 0

21) 1

2

22) 1

23) (a) dy

dx = -

1 + y2

1 + x2 ; (b) 5x + 2y = 9

24) arctan(ex) + C

25) 2

3 tan-1(

3x

2 ) + C

26) 1

4 (arcsin(2x))2 + C

27) e2

28) 2

29) diverges

6

Answer Key Testname: MATH 2414 HCCS FINAL EXAM REVIEW

30) Converges

31) Converges

32) 8 ≤ x < 10

33) x - 1

6 x3 +

1

120 x5

34) P3(x) = 1

2 -

1

4 x +

1

8 x2 -

1

16 x3 +

24

(z + 2)5 ∙

x4

4! for some z between 0 and x.

35) ∞

n = 0

(-1)nx2n + 2

n + 1 ∑

36) y = 5x

37) ( 3 3

2 , -

3

2 )

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

38) r2 = 2

(cos θ)(sin θ)

39) (x2 + y2) 3

= 36x2y2

7

Answer Key Testname: MATH 2414 HCCS FINAL EXAM REVIEW

40) ( x

4 ) 2

+ ( y

7 ) 2

= 1

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

8