Practice Final - Ohio State University

Math 161.01 - Accelerated Calculus I 108 min

Practice Final

December 7, 2010

? Calculators are allowed as long as they have no symbolic integration capbility (TI-84 and comparable are ok)

? Remember to CIRCLE YOUR FINAL ANSWER.

Useful facts:

?

Volume of a sphere V

=

4 3

r3

?

Volume of a cone V

=

1 3

r2h

? F = ma (force is mass times acceleration)

? m = V d (mass is volume times density)

? acceleration of gravity is -9.8 m/s2

?

sin A sin B

=

1 2

[cos(A

-

B)

-

cos(A

+

B)]

?

sin A cos B

=

1 2

[sin(A

-

B)

+

sin(A

+

B)]

?

cos A cos B =

1 2

[cos(A

-

B

)

+

cos(A

+

B)]

?

sin2 x =

1 2

(1

-

cos

2x)

?

cos2 x =

1 2

(1

+

cos

2x)

1

1. (10 points) The derivative of a continuous function h is pictured below.

-5 -4 -3 -2 -1 0 1 2 3 4 5

h

Sketch a continuous function h whose derivative could be the given graph for h .

-5 -4 -3 -2 -1 0

h

1 23 4 5

2

2. Evaluate the following limits using any technique you like.

(a) lim x + 2 - x

x

x+2+ x

lim x + 2 - x = lim ( x + 2 - x) ?

x

x

x+2+ x

x+2-x

= lim

x x + 2 + x

2

= lim

x x + 2 + x

=0

(ln x)2 (b) lim

x ln(ln x)

(ln x)2

2(ln x)x-1

lim

x

ln(ln x)

=

lim

x

(ln x)-1x-1

= lim 2(ln x)2 x

=

1

(c) lim 1 + e-x x x

lim

x

1 + e-x

1 x

=

lim

ex-1 ln(1+e-x)

x

= elimx x-1 (ln 1+e-x)

= e0?ln 1

=1

3. Evaluate the following derivatives.

(a) d ex + xe + arctan(3 - 84) dx d ex + xe + arctan(3 - 84) = ex + ex(e-1) dx

(b) d ex sin(x2) dx

d ex sin(x2) = ex sin(x2) + 2xex cos(x2) dx

dx (c)

dx x + 1

dx

x+1-x

1

=

=

dx x + 1

(x + 1)2 (x + 1)2

3

d

xx

(d)

dx (x + 1)3 sin x

Let

y

=

. xx

(x+1)3 sin x

Then

ln y = x ln x - 3 ln(x + 1) - ln(sin x)

Hence, Hence,

y

3 cos x

= ln x - 1 -

-.

y

x + 1 sin x

xx

3

y=

ln x - 1 -

- cot x .

(x + 1)3 sin x

x+1

4. Evaluate the following indefinite integrals.

(a) x arctan x dx

u=arctan x,

du=

dx x2 +1

,

dv=x dx,

v=

x2 2

x2 arctan x 1

x2

x arctan x dx =

-

dx

2

2 x2 + 1

x2 arctan x 1

1

=

- 1-

dx

2

2

x2 + 1

x2 arctan x x arctan x

=

-+

+C

2

2

2

(b) sin17 x cos3 x dx

u=sin x, du=cos x dx

sin17 x cos3 x dx = sin17 x(1 - sin2 x) cos x dx

= u17(1 - u2) du

= u17 - u19 du

u18 u20 = - +C

18 20

sin18 x sin20 x

=

-

+C

18

20

4

sec3 x

(c)

dx

tan x

sec3 x dx =

tan x = =

sec x(1 + tan2 x) dx

tan x

sec x

sec x tan2 x

dx +

dx

tan x

tan x

csc x dx + sec x tan x dx

= ln | csc x - cot x| + sec x + C

(d) sin4 x dx

sin4 x dx =

(1 - cos 2x)2

22

dx

1 =

1 - 2 cos 2x + cos2 2x dx

4

1

1 + cos 4x

= 1 - 2 cos 2x +

dx

4

2

13

cos 4x

=

- 2 cos 2x +

dx

42

2

1 3x

sin 4x

=

- sin 2x +

+C

42

8

3x sin 2x sin 4x

=-

+

+C

8

4

32

(e) cos 2x cos 9x dx

1 cos 2x cos 9x dx =

2 1 = 2 1 = 2

cos(2x - 9x) + cos(2x + 9x) dx

cos(-7x) + cos(11x) dx

sin(-7x) sin(11x)

+

+C

-7

11

sin(-7x) sin(11x)

=-

+

+C

14

22

2x

(f)

dx

x2 - 9

u=x2-9, du=2x dx

5

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