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If $$f'(x) = \sin\left(\frac{\pi}{2}e^x\right)$$ and $f(0) = 1$ then $f(2) =?$

I'm currently studying for my calculus exam and came across this multiple choice question. I have tried to do $u$ substitution to get $f(x) + c$. And then to just plug in the values. I have been dwelling on this for half an hour now either I'm doing it right and getting the answer wrong I have the wrong approach. Would someone please be kind enough to show a step by step guide for questions like these in general ( so its useful for other people as well).

The choices are:

  • A $-1.819$
  • B $-0.843$
  • C $-0.819 $
  • D $0.157$
  • E $1.157 $
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That does not have an obvious integral so I suspect there is an error in transcription –  Henry Feb 25 '12 at 13:53
    
@Henry: You are right I think as the OP used "x" for times and the independent variable. I think this should be $\sin\left(\frac{\pi}{2}e^x\right)$. OP should correct. –  Jon Feb 25 '12 at 14:16
    
@Jon: Even then, you need to use the sine integral, which is not suitable for multiple choice questions. –  Henry Feb 25 '12 at 14:21
    
@Henry: You are right. Indeed, $\int dx\sin\left(\frac{\pi}{2}e^x\right)$. I put $y=e^x$ and so $\frac{dy}{y}=dx$ and so we are left with sine integral. Unless this is homework requiring some numerical evaluation. –  Jon Feb 25 '12 at 14:31
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Since the set of $x$ with $f'(x) < 0$, has total length $<1$, and $f'\ge -1$ there, the answer is positive. That rules out three choices. How close to zero can the answer be? Not 0.157... That leaves 1.157. –  GEdgar Feb 25 '12 at 17:54

1 Answer 1

Here is a picture.

enter image description here

The function $f'$ is in red. Consider another function $g$ with $g \le f'$, seen in green. The points on the $x$-axis are 0.6, 1.4, 1.7. The negative parts are $-1$, the positive parts are piecewise linear, with peaks where the curve is maximum, namely at $x=0$ and at $x=\log 5$. If we compute $1+\int_0^2 g(x)dx$ it is easy, two rectangles of known size, two triangles of height 1 and known widths, plus the 1. The answer is 0.35. But this is smaller than the value $1+\int_0^2 f'(x)dx$. The only choice bigger than 0.35 is, therefore, the answer.

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