M121 Exam 2.

Show your work. You will be graded out of 100 points; there are 30 possible points in part I and 90 possible in part II.

Part I. 5 each

  1. If b; use the quotient rule:

a)

b)

c)

d)

2. Suppose you know that . Find if Ans.: “b”:

a)

b)

c)

d)

  1. If and find where : Ans.: “a”

a)

b)

c)

e)none of the above

4-6: Let

4. s is at its maximum when: ANS: “c”:

a)

b)

c)

d)none of the above

  1. If a(t) = v’(t) then which ONE of the following is true: ANS: “d”;

a)a varies with t.

b)a is proportional to s

c)a is proportional to

d)a is constant

  1. If then ONE of the following is true: ANS: “b”;

a)v is constant

b)v is positive at t=0 and negative at

c)v is proportional to

d)v is positive for all t.

Part II. (15 each)

1. Suppose you know that and and .

a)Estimate .

b)Is your estimate likely to be too large or two small? Why?

This is likely to be too small as f is concave up (second derivative is positive) and therefore the tangent line lies below the graph.

2. On what interval(s) is the function increasing? Assume

f increases where

  1. A spherical balloon is being inflated. Recall that surface area .Find when and give a brief (one sentence) explanation of what this quantity is. For a little bit of extra credit, describe geometrically WHY a certain well known formula appears in this calculation.

this is the rate of change of surface area with respect to radius. When r =2, you get .

Extra credit: note that this is = 4*(circumference). To see this: take the sphere, cut it along its “equator” into 2 hemispheres. Then slice a hemisphere along a “great circle” to get a “spherical triangle” with base equal to and sides equal to . Adding an increment of “r” to each side gives that increment times . But there are 2 hemispheres, hence the value is .

  1. A physics formula for force is the following: where m is mass and v is velocity.

a)If a truck has mass m = 4000 kg, is moving at v = 28 m/sec. , accelerating at 2 m/(sec)*(sec) and losing mass at the rate of .5kg/sec due to a sand leak and fuel consumption, find F .(recall, acceleration is the derivative of v)

b)You’ve probably seen the formula where . Can you think of when this formula is the same as the formula given in this problem?

For part “b”, note that the term is missing. This is because m is often assumed to be constant; hence .

5. Suppose

a)Is f continuous at x = 0? Why or why not?

b)Is f differentiable at x = 0? Why or why not? (That is, does exist?)

a)Yes,

b)Yes, , hence the derivative not only exists, but is continuous.

6. How many tangent lines to the curve pass through the point (1, 2) ? At which points do these tangent lines touch the curve (give the x-value)?

Solution: this involves knowing that the slope of the line tangent to the graph of “y” is given by the derivative. which should be equated to the slope of the lines which are going to be tangent to the graph and run through the point (1, 2). These slopes, which can be thought of as “rise over run” can be given by So we have two lines. By the way, we used the quadratic formula to solve for “x”.

This problem was worked in each class section!