MATH 105 FINAL EXAM April 13, 2010

Name:

Your grade is based on correctness, completeness, and clarity on each

exercise. You may use a calculator, but no notes, books, or other students.

Good luck!

1.) (15 pts.) Compute the following derivatives, using any valid techniques you like. (Be

sure to show enough steps so that it is clear which techniques you are using.)

a.) (5 pts.) f(x) = √π+extan x

b.) (5 pts.) p(t) = 1

t+1

1 + t2

c.) (5 pts.) y=xcos x

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2.) (15 pts.) Consider the integral Z2

−1

(−x+ 1) dx.

a.) (5 pts.) Interpret the integral as “area under a curve”. Draw the graph with area

shaded in, and use geometry to compute the signed area.

b.) (5 pts.) Evaluate the integral using the Fundamental Theorem of Calculus.

c.) (5 pts.) Are your answers the same or diﬀerent? Explain why your answers should, or

should not, be the same.

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3.) (15 pts.) Consider the integral Z2

1

1

x2dx.

a.) (5 pts.) Approximate the area under the curve using the a trapezoid sum with two

equal subintervals. In your approximation, use at least two digits after the decimal

point.

b.) (5 pts.) Evaluate the integral using the Fundamental Theorem of Calculus.

c.) (5 pts.) Are your answers the same or diﬀerent? Explain why your answers should, or

should not, be the same.

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4.) (20 pts.) Use the graph of f(x), below, to complete the following questions.

-5

-4

-3

-2

-1

1

2

3

4

5

x

-5

-4

-3

-2

-1

1

2

3

4

5

fHxL

-5

-4

-3

-2

-1

1

2

3

4

5

x

-5

-4

-3

-2

-1

1

2

3

4

5

f¢HxL

a.) At which x-value(s)

i.) (2 pts.) does F, an antiderivative of f, change from concave up to concave down?

ii.) (2 pts.) does F, an antiderivative of f, have a stationary point?

iii.) (2 pts.) does F, an antiderivative of f, have a local maximum?

iv.) (2 pts.) is fgreatest?

v.) (2 pts.) is f0greatest?

vi.) (2 pts.) is F, an antiderivative of f, greatest?

b.) (2 pts.) What kind of symmetry does fhave?

c.) (4 pts.) Sketch the derivative f0(x) on the given set of axes.

d.) (2 pts.) What kind of symmetry does f0have?

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