Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $
share|cite|improve this question
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
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.