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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 transcription2012-02-25
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    @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.2012-02-25
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    @Jon: Even then, you need to use the sine integral, which is not suitable for multiple choice questions.2012-02-25
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    @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.2012-02-25
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    Is $f(2)=1+ \int_0^2 \sin\left(\frac{\pi}{2}xe^x\right)$ among the choices?2012-02-25
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    @John: When you ask about a multiple choice question, you should supply the choices, since deciding between supplied alternatives is alternatives-dependent.2012-02-25
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    Thanks a lot for your response guys, the choices are as follows: A -1.819 B -0.843 C -0.819 D 0.157 E 1.157 And there is no additional x inside the sin bracket, it should say f′(x)=sin((π/2)e^x)2012-02-25
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    @John: Edit your question removing x from the sine argument and add choices.2012-02-25
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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 1.157.2012-02-25
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    Hi @GEdgar, you're a professional mathematician, right? Unfortunately your argument seems a bit too succint for me... Could you elaborate on that please? Eg. "total length < 1" is not clear to me at all... I believe it's something like "since $(e^x)'$ is strictly positive, ..." Moreover I don't see why 0.157 is too close to zero. BTW, here is [f'(x) plotted by Google](https://www.google.com/search?q=sin(pi%2F2*e%5Ex)+from+-0.5+to+2.5)2012-02-25
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    Let $$\int_{0}^{2} f'(x) dx = A$$ where $A =$ area under $f'(x)$ between $0 \le x \le 2$. But $$\int_0^2 f'(x) dx = f(2) - f(0).$$ So $$f(2) = A +1$$ Show that $A$ is positive for $0 \le x\le 2$, and the answer will be $1.157$. But I don't know how to show that!2012-02-25
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    @nodakai: That plot has $e^x$ in the denominator, unlike the current statement of the question. Which is it?2012-02-25
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    @GEdgar: Oops, "*" disappeared somewhere while I copy&pasted the URL... [This is a corrected version](https://www.google.com/search?q=sin(%28pi%2F2%29e%5Ex)+from+-0.5+to+2.5)2012-02-25

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