[PDF] CALCULUS I, Final Exam 1 - UAB

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CALCULUS I, Final Exam 8 3 12 points Graph the function y= f(x) = x2 x2 1 Find xand y-intercepts, hor-izontal and vertical asymptotes, all critical numbers, intervals of in-/de-creasing, local/absolute max/min Draw your graph on the next page

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CALCULUS I, Final Exam 1

MA 125 CALCULUS I

Final Exam, December 10, 2014

Name (Print last name rst):::::::::::::::::::::::::::::::::::::::::::

Show all your work, justify and simplify your answer!No partial credit will be given for the answer only!

PART I

You must simplify your answer when possible but you don't need to com- pute numbers: e

6sin(12=5) +8 is a ne answer.

All problems in Part I are 4 points each.

1. Use the denitionof the derivative to show that the derivative of the function y=f(x) =x2isf0(x) = 2x. 2.

Find the deriv ativef0(x) iff(x) =x2sin(x).

3.

Find the deriv ativef0(x) iff(x) = ln(x3+x2+ 1).

CALCULUS I, Final Exam 2

4.

Find the deriv ativef0(x) iff(x) =x3+1x

31.
5.

Find the an ti-derivative

Rx2(1 +px)dx.

6.

Find the an ti-derivative

Rsin6(x)cos(x)dx.

7.

Find the an ti-derivative

Rx3px

4+ 5dx.

CALCULUS I, Final Exam 3

8.

So lveln( x2+ 1) = 5.

9.

If F(x) =Z

x 2 sin(t2+ 1)dt, ndF0(x). 10. If oil leaks from a w ellat the rate of e5t(m3=s), how much oil will leak in the rst minute? (If you use your calculator to compute it is OK if you give an approximate answer.)

CALCULUS I, Final Exam 4

11.

App roximate

Z 4 11x dxusing a Riemann sum withn= 3 terms and the midpoint rule. What does this number have to do with ln(4)? 12. T hev elocityof a part icleis giv enb yv(t) =t2+ 1 (m=s) . (a)

Fin dthe acceleration a(2) of the particle,

(b) Ho wfar do esth eparticle tra velin the rst 5 seconds?

CALCULUS I, Final Exam 5

13. Gi venthe graph of the function f(x) below answer the following questions.xy

123451020304050y=f(x)(a)Is f(x) one-to-one? Explain!!

(b)

Use the graph to appro ximatef1(20).

(c)

Use the graph to appro ximate( f1)0(20).

CALCULUS I, Final Exam 6

PART II1.9 points.Find all local/absloute maxima/minima of the function f(x) = (2x+ 1)3(1x)5on the real line (1;1).

CALCULUS I, Final Exam 7

2.9 points.LetS(t) be the function which species the distance (in km) from a

runner to the start line at timet(in hours) of a race. The graph ofS(t) is given below:t(h)y(km)123451234 S(t)Use the graph to give approximate answers to the following problems. (a)

Whe nw asthe runner runn ingthe fastest?

(b)

W hathapp enedb etweentimes 3 and 4?

(c)

What is the mean ingof S1(1)?

CALCULUS I, Final Exam 8

3.12 points.Graph the functiony=f(x) =x2x

21. Findxandy-intercepts, hor-

izontal and vertical asymptotes, all critical numbers, intervals of in-/de-creasing, local/absolute max/min

Draw your graph on the next page.

CALCULUS I, Final Exam 9

xy

CALCULUS I, Final Exam 10

4.9 points.EvaluateZ

1 0x

2(5x)dx.

CALCULUS I, Final Exam 11

5.9 points.Find the dimensions of a can (i.e. a cylinder) of radiusr, heighth

and volume 1(m3) with minimal surface area. [Hint: the volumeV=r2hand the surface areaS= 2rh+ 2r2.]

CALCULUS I, Final Exam 12

Scratch paper

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