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Optimizations, Limits, Concavity and Derivatives

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Can you help me with the following questions of this Assignment:
Q. 7, 8, 11, 12, 13

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This solution is comprised of a detailed explanation to answer calculus question.

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11. Since f '(x) = 2x+1
f(x) = ∫ (2x+1) dx
= x² +x + C
So f(1)= 1+1+c = 2+c
On the other hand it is given that f(1)=6 .
Therefore we get 2+c = 6, so c=4

Hence, f(x) = x² +x+4

12. f"(x) = 2 e^t +3sint

Therefore f ' (t) = ∫ f"(t) dt
= ∫ 2 e^t dt + ∫ 3sint dt
= 2 e^t -3 cost + C

That gives: f(t) = ∫ 2 e^t dt- ∫ 3 cost dt+ ∫ C dt
= 2e^t +3sint +Ct+D

So f(0) = 2e^0 + 3 sin0 + C*0+D
=2 ...

Solution provided by:
Chris Busu, PhD (IP)
Education
  • BSc, University of Bucharest
  • MSc, Ovidius
  • MSc, Stony Brook
  • PhD (IP), Stony Brook
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