6.2. THE SQUEEZE THEOREM AND THE MEAN VALUE THEOREM 209
Problem 6.28 Find the value of c for the mean value theorem for the following functions and
intervals Œa; b.
1. f .x/ D 4 x
2
on [-2,1]
2. g.x/ D x
3
on [-
1,1]
3. h.x/ D ln.x/ on
[1,4]
4. r.x/ D
p
x on
[0,9]
5. s.x/ D sin.x/ on
Œ0;
2
6. q.x/ D tan
1
.x/
on [0,1]
Problem 6.29 For each part of Problem 6.28, carefully sketch the situation as in the examples
in this section.
Problem 6.30 True or false:
lim
x!0
x sin
1
x
is zero. Explain your answer.
Problem 6.31 True or false:
lim
x!0
3
p
x cos
1
x
is zero. Explain your answer.
Problem 6.32 True or false:
lim
x!0
p
x sin
1
x
exists. Explain your answer.
Problem 6.33 Construct an example of f .x/, a, and b so that the point c satisfying the mean
value theorem is not unique.
Problem 6.34 Using the mean value theorem, prove that
jsin.u/ sin.v/j ju vj
for any u; v.
Problem 6.35 Find two values a; b such that the secant line of y D e
x
has a slope of m D 4.
Having done this, find the formula of the tangent line with the same slope.
210 6. LIMITS AND CONTINUITY: THE DETAILS
Problem 6.36 Find two values a; b such that the secant line of y D ln.x/ has a slope of m D 2.
Having done this, find the formula of the tangent line with the same slope.
Problem 6.37 Find the maximum slope of any secant line of
y D cos.x/
Problem 6.38 Suppose we take a car trip and have a digital logger record our velocity through-
out the trip. Since the car is a mechanical device in the real world, we know that velocity as a
function of time is a continuous function. If the trip took three hours, and we went 140 km,
what is the slope at the mean value theorem point c for the logged function?
Problem 6.39 Show that there is a value x D a so that the tangent line to y D cos.x/ at
.a; cos.a// goes through the point .4; 3/.
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