2007fall1amti.pdf

7/30/2019 2007Fall1aMTI.pdf

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Name: ID#:

Midterm I

Math 1aIntroduction to Calculus

October 24, 2007

Please check your section:

0.0 MWF9 Sam Isaacson 4.0 TTH10 Samit Dasgupta 1.0 MWF10 Ji Oon Lee 4.1 TTH10 Jeechul Woo 1.1 MWF10 Matthew Leingang 5.0 TTH11:30 Rehana Patel 1.2 MWF10 Stewart Wilcox 2.0 MWF11 Si Li 2.1 MWF11 David Smyth 3.0 MWF12 John Duncan

Rules:

This is a two-hour exam. Calculators are not allowed.

Unless otherwise stated, show all ofyour work. Full credit may not begiven for an answer alone.

You may use the backs of the pages orthe extra pages for scratch work. Donot unstaple or remove pages as they canbe lost in the grading process.

Please do not put your name on anypage besides the first page. If you like,

you may put your ID number on thetop of each page you write on.

Please, please, please dont cheat.

Hints:

Read the entire exam to scan for ob-vious typos or questions you mighthave.

Budget your time so that you dontrun out.

Problems may stretch across severalpages.

Relax and do well!

Good luck!


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Problem Possible PointsNumber Points Earned

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

10 10

Total 100


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1 Math 1a Midterm I October 24, 2007 1

1. (10 Points)

(a) (2 points) Let f be a function and a a point in the domain of f. State the definition off(a), the derivative of the function at a.

(b) (8 points) Let f(x) = x2 1. Find an equation of the tangent line to the graph of f(x)at the point (1, 0) by using the definition of the derivative. No credit will be given for usingthe Power or any other Rules.

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2. (10 Points) Let f be given by

f(x) =

2x + 3 ifx 1

x2 + x + 1 ifx > 1

Justify your answers to both of these:

(i) Is f continuous at 1?

(ii) Is f differentiable at 1?

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3. (10 Points) The population of a city is measured at two year intervals and is given asfollows:

Year 1984 1986 1988 1990 1992 1994Population 265 290 324 358 395 437

Let P(t) be the population at any time t.

(a) Estimate the rate of population growth in 1988, 1990, and 1992.

(b) Based on your answers in (a), do you expect the graph of P(t) to be concave up ordown?

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4. (10 Points) The graphs of two functions f and g, defined on (, 3), are given below.Use them to evaluate each of the following limits, if they exist. If a limit does not exist, oris , say so and explain why.

f

2 3

1

2

3

g

2 3

1

2

3

4

(i) (2 points) limx2

f(x)

(ii) (3 points) limx2

[f(x) + g(x)]

(iii) (2 points) limx [f(x) g(x)]

(iv) (3 points) limx0f(x)

g(x)

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5. (10 Points) Let

f(x) =

x2 + 1

x + 3

Compute the following limits, and explain your answers.

(a) limx f(x)

(b) limx f(x)

(c) limx3+

f(x)

(d) limx3

f(x)

(e) Which of these could be the graph of f? Explain.

0

(A)

0

(B)

0

(C)

0

(D)

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6. (10 Points) Derivative shortcuts are allowed on this problem. Find the following derivatives.

(i) (3 points) f(x), where f(x) = x3 + 3x2 4

(ii) (2 points) f(x), where f is as above

(iii) (3 points)d

dx(3ex)

(iv) (2 points)dy

dx, where y = 14.

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7. (10 Points) Derivative shortcuts are allowed on this problem. The height above ground of aball at time t is given by the function

s(t) = 16t2 + 40t + 10

(i) (4 points) Compute the balls velocity at t = 1. Is the ball going up or down?

(ii) (2 points) Compute the balls velocity at t = 2. Is the ball going up or down?

(iii) (4 points) Is the graph of the position function concave up or down?

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8. (10 Points) In this problem, we will show that every cubic equation has a real solution.More precisely, well show that if f is a function of the form

f(x) = x3 + ax2 + bx + c

(here a, b, and c are constants), then the graph crosses the x-axis.

(a) (2 points) Explain why such a function f is continuous.

(b) (5 points) Show that limx f(x)x3 = 1 and limx f(x)x3 = 1.

(c) (3 points) It follows from (b) that f has a positive value at some point and a negativevalue at another point. You dont have to show that. But use this fact to show that fhas to achieve the value zero at some point. Give the name of any important theoremsyou use.

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9. (10 Points) Match the graphs of each of the five functions below with the graphs of theirderivatives. No justification is necessary.

(i) (ii) (iii)

(iv) (v)

Derivatives:

(A) (B) (C) (D)

(E) (F) (G) (H)

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10. (10 Points) A function f(x) satisfies

x + 1 x2 f(x) x + 1 + x2

for all x. In particular, f(0) = 1. In each of the following, give the name of any importanttheorem you use.

(a) (3 points) Show that f is continuous at 0. That is, show that

limx0

f(x) = f(0).

(b) (5 points) Evaluate directly the limit

limh0+

f(h) f(0)h

Hint. You may be tempted to think that this is the derivative of f at zero, then takethe derivative of the inequality above and evaluate at zero. This is specious (wrong)

because (1) we dont yet know if f is differentiable at zero and (2) differentiating aninequality doesnt necessarily preserve the inequality. Instead, simplify and take thelimit.

(c) (2 points) Its also true that limh0

f(h) f(0)h

exists and is equal to the limit found in

(b) (you dont have to show that). Is f differentiable at 0?

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2007fall1amti.pdf

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