Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.
- (20 points)
The graph of a function \(f\) is given in each part below.
Copy this graph and then sketch the graph of \(f'\) on the same axes.
- (15 points) Sketch the graph of a differentiable function \(g\) for which \(g(0)=g'(0)=0\),
\(g'(-1)=-1\), \(g'(1)=3\), and \(g'(2)=1\).
- (16 points)
Compute the following derivatives:
- \(\displaystyle \frac{d}{dx} \left[3\sin(x)\right]=\)
- \(\displaystyle \frac{d}{dx} \left[x\sin(3)\right]=\)
- \(\displaystyle \frac{d}{dx} \left[3\sin(3)\right]=\)
- \(\displaystyle \frac{d}{dx} \left[x\sin(x)\right]=\)
- \(\displaystyle \frac{d}{dx}\left[\frac{\sin(x)}{3}\right]=\)
- \(\displaystyle \frac{d}{dx}\left[\frac{\sin(3)}{x}\right]=\)
- \(\displaystyle \frac{d}{dx}\left[\frac{\sin(3)}{3}\right]=\)
- \(\displaystyle \frac{d}{dx}\left[\frac{\sin(x)}{x}\right]=\)
- (30 points)
Compute the following derivatives:
- \(\displaystyle \frac{d}{dx} \left[(9\cos(x)+x^8+x^5+3)\sin(x)\right]=\)
- \(\displaystyle y=\frac{x^3+5x}{\sin(x)} \Rightarrow \frac{dy}{dx}=\)
- \(\displaystyle \frac{d}{dx} \left[\frac{\sin(x)\cos(x)}{x^3+x}\right]=\)
- (19 points)
- Use the Product Rule twice to prove that if \(f\), \(g\), and \(h\) are differentiable, then \((fgh)'=f'gh+fg'h+fgh'\).
- Find the corresponding formula for \(\displaystyle\left(\frac{f}{gh}\right)'\).
- Find the corresponding formula for \(\displaystyle\left(fg\right)''\).
Last modified: Thu Jan 27 20:00:03 UTC 2022