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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 *+ *ex *tan *x
*

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

*t
*+

1

1 + *t*2

c.) (5 pts.) *y *= *x*cos *x
*

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2.) (15 pts.) Consider the integral

∫ 2
*−*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

∫ 2 1

1

*x*2
*dx*.

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
f HxL
**

**-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 *f *greatest?

v.) (2 pts.) is *f ′ *greatest?

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

b.) (2 pts.) What kind of symmetry does *f *have?

c.) (4 pts.) Sketch the derivative *f ′*(*x*) on the given set of axes.

d.) (2 pts.) What kind of symmetry does *f ′ *have?

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5.) (15 pts.) Use the graph of *g*(*x*), below, to complete the following questions.

**1 2 3 4 5
x
**

**1
**

**2
**

**3
**

**4
**

**5
**

*g*H*x*L

a.) (3 pts.) What is lim
*x→*2*−
*

*g*(*x*)?

b.) (3 pts.) What is lim
*x→*2+

*g*(*x*)?

c.) (3 pts.) What is lim
*x→*2

*g*(*x*)?

d.) (3 pts.) What is *g*(2)?

e.) (3 pts.) What are the hypotheses of the Mean Value Theorem? Does *g*(*x*) satisfy those
hypotheses for *x*-values in the interval [0*, *5]?

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6.) (20 pts.) A 13-ft ladder is leaning against a vertical wall when the foot of the ladder begins to slide away from the wall at a rate of 0.5 ft/s. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 5 ft from the wall?

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