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1. Find f 0 (x) where f (x) = 3x .
2. Find f 0 (x) where f (x) = (2x)1/x .
x
3. Find f 0 (x) where f (x) = (x)e .
4. Find f 0 (x) where f (x) = (x + 1)1−x .
´
5. Evaluate (2x + 7)5 dx.
´
6. Evaluate (ax + b)k dx for all k ∈ R.
7. Find f such that f 0 (x) = e5x .
8. Find f such that f 0 (x) =
2
x3/2
and f (1) = 5.
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9. Find f such that f 00 (x) =
1
,
x2
f (2) = 0 and f (1) = 1.
10. Find the area enclosed by the curves y = x2 − 4 and y = 4 − x2 .
11. Find the area under the curve y(x) = x2 − 6x + 5 from x = −1 to x = 3.
12. (Textbook, p. 289, ex. 13) The population of a certain country is growing
exponentially. The total population (in millions) in t years is given by the
function P (t). Match each of the following answers with its corresponding
question.
Answers:
(a) Solve P (t) = 2 for t.
(e) y 0 = ky
(b) P (2)
(f) Solve P (t) = 2P (0) for t
(c) P 0 (2)
(g) P0 ekt , k > 0
(d) Solve P 0 (t) = 2 for t
(h) P(0)
Questions:
(I) How fast will be the population growing in 2 years?
(II) Give the general formula of the function P (t).
(III) How long will it take for the current population to double?
(IV) What will be the size of the population in 2 years?
(V) What is the initial size of the population?
(VI) When will be the size of the population 2 million?
(VII) When will be the population growing at the rate of 2 million people per
year?
(VIII) Give the differential equation satisfied by P (t).
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