Hello, Susie38!
Here are parts (b) and (c).
 \:= \:\begin{Bmatrix}\frac{2}{\pi}x & 0 \leq x \leq \frac{\pi}{2} \\ 2\sin x & \frac{\pi}{2} < x \leq \frac{3\pi}{2} \\ 1 & \frac{3\pi}{2} < x \leq 2\pi\end{Bmatrix})
a. Find a continuous function,
)
, satisfying
 = \int^x_0 f(t)\,dt)
b. Give the values of
on
![[0,2\pi]](http://latex.codecogs.com/png.latex?[0,2\pi])
for which
is not differentiable.
c. Compute:
\,dx)
On ![\left[0,\,\frac{\pi}{2}\right]\]](http://latex.codecogs.com/png.latex?\left[0,\,\frac{\pi}{2}\right]\])
, the function is a straight line.
On ![\left(\frac{\pi}{2},\,\frac{3\pi}{2}\right]](http://latex.codecogs.com/png.latex?\left(\frac{\pi}{2},\,\frac{3\pi}{2}\right])
, the function is a sine curve.
On ![\left(\frac{3\pi}{2},\,2\pi\right]](http://latex.codecogs.com/png.latex?\left(\frac{3\pi}{2},\,2\pi\right])
, the function is a horizontal line. Code:
|
1 + ** o * * * *
| * : * : :
| * : * : :
| * : * : :
- * - - - + - - - * - - - + - - - + -
| π/2 2π* 3π/2 2π
| * :
| * :
-1 + **
| (b) is not differentiable at 
and 
It has two different slopes at
It is discontinuous at
(c) No calculus needed!
The triangle has base 
and height 
. . . Its area is: (1)\:=\; \frac{\pi}{4})
The sine curve has a region above and a region below the x-axis.
. . . The two regions have equal area and will cancel out.
Its net area is: 
The rectangle has base and height
. . . Its area is: (1)\:=\: \frac{\pi}{2})
The value of the integral is the sum of these areas:
. . . 
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