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in which a radioactive substance is decaying at the rate of 13:38 the constant solution and determine whether the solution
5% but the substance is being replenished at the constant x(t) will increase or decrease. Based on these conclusions,
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rate of 2. Find the value of x(t) for which x (t) = 0. Pick conjecture the value of lim x(t), the limiting amount of ra-
t→∞
various starting values of x(0) less than and greater than dioactive substance in the experiment.

Review Exercises

WRITING EXERCISES In exercises 1 and 2, find the linear approximation to f(x) at x .
0
The following list includes terms that are defined and theorems 1. f(x) = e , x = 0 2. f(x) = √ x + 3, x = 1
3x
2
0
0
that are stated in this chapter. For each term or theorem, (1) give ............................................................
a precise definition or statement, (2) state in general terms what
it means and (3) describe the types of problems with which it is
associated. In exercises 3 and 4, use a linear approximation to estimate the
quantity.
Linear approximation Newton’s method Critical number √
Absolute extremum Local extremum First Derivative Test 3. 3 7.96 4. sin 3
Inflection points Concavity Second Derivative ............................................................
Marginal cost Current Test
l’Hôpital’s Rule Extreme Value Related rates
Theorem Fermat’s Theorem In exercises 5 and 6, use Newton’s method to find an approxi-
mate root.
3
3
5. x + 5x − 1 = 0 6. x = e −x
............................................................
TRUE OR FALSE
State whether each statement is true or false and briefly explain
why. If the statement is false, try to “fix it” by modifying the given 7. Explain why, in general, if y = f(x) has an inflection point
statement to make a new statement that is true. at x = a and does not have an inflection point at x = b, then
the linear approximation of f(x) at x = a will tend to be more
1. Linear approximations give good approximations of func- accurate for a larger set of x’s than the linear approximation
tion values for x’s close to the point of tangency. of f(x) at x = b.
2. The closer the initial guess is to the solution, the faster 8. Show that the approximation 1 ≈ 1 + x is valid for
Newton’s method converges. (1 − x)
“small” x.
3. L’Hôpital’s Rule states that the limit of the derivative equals
the limit of the function.
In exercises 9–16, find the limit.
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4. If there is a maximum of f(x) at x = a, then f (a) = 0. 3
5. An absolute extremum must occur at either a critical num- 9. lim x − 1 10. lim 2 sin x
x→1 x − 1
x→0 x + 3x
2
ber or an endpoint. 2x
2 −3x
′
′
6. If f (x) > 0 for x < a and f (x) < 0 for x > a, then f(a)isa 11. lim 4 e 12. lim(x e )
x→∞ x + 2
x→∞
local maximum. √
x 2 −4
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7. If f (a) = 0, then y = f(x) has an inflection point at x = a. 13. lim | x + 1 | | 14. lim x ln(1 + 1∕x)
|
x→2 + | | x→∞
| x − 2 |
8. If there is a vertical asymptote at x = a, then either tan −1 x
lim f(x) =∞ or lim f(x) = −∞. 15. lim (tan x ln x) 16. lim −1
x→a + x→a + x→0 + x→0 sin x
9. In a maximization problem, if f has only one critical num- ............................................................
ber, then it is the maximum.
10. If the population p(t) has a maximum growth rate at t = a, In exercises 17–26, do the following by hand. (a) Find all critical
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then p (a) = 0. numbers, (b) identify all intervals of increase and decrease, (c)
determine whether each critical number represents a local max-
dg
′
′
11. If f (a) = 2 and g (a) = 4, then = 2 and g is increasing imum, local minimum or neither, (d) determine all intervals of
df concavity and (e) find all inflection points. Copyright © McGraw-Hill Education
twice as fast as f.
4
17. f(x) = x + 3x − 9x 18. f(x) = x − 4x + 1
3
2
316 | Lesson 4-9 | Rates of Change in Economics and the Sciences