70 Part I: The Questions
489. Apply the mean value theorem to the
function f
 
(x) = x
1/3
on the interval [8, 9]
to find bounds for the value of 9
3
.
Relating Velocity and Position
490–492 Use the position function s(t) to find the
velocity and acceleration at the given value of t.
Recall that velocity is the change in position with
respect to time and acceleration is the change in
velocity with respect to time.
490. s(t) = t
2
– 8t + 4 at t = 5
491. s(t) = 2 sin t – cos t at t = π
2
492. s t
t
t
( ) = +
2
1
2
at t = 1
Finding Velocity and Speed
493– 497 Solve the given question related to speed
or velocity. Recall that velocity is the change in
position with respect to time.
493. A mass on a spring vibrates horizontally
with an equation of motion given by x(t) =
8 sin(2t), where x is measured in feet and t is
measured in seconds. Is the spring stretching or compressing at t = π
3
? What is the
speed of the spring at that time?
Using the Mean Value
Theorem
484– 486 Verify that the given function satisfies the
hypotheses of the mean value theorem. Then find
all numbers c that satisfy the conclusion of the mean
value theorem.
484. f
 
(x) = x
3
+ 3x – 1, [0, 2]
485. f x
x
( ) = 2
3
, [0, 1]
486. f x
x
x
( ) = + 2
, [1, 4]
Applying the Mean Value
Theorem to Solve Problems
487– 489 Solve the problem related to the mean
value theorem.
487. If f
 
(1) = 12 and f
 
'(x) ≥ 3 for 1 ≤ x ≤ 5, what is
the smallest possible value of f
 
(5)? Assume
that f satisfies the hypothesis of the mean
value theorem.
488. Suppose that 2 ≤ f
 
'(x) ≤ 6 for all values of x.
What are the strictest bounds you can put
on the value of f
 
(8) – f
 
(4)? Assume that f is
differentiable for all x.