Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.
- (20 points) Use implicit differentiation to find an equation for the tangent
line to the curve defined by
\( \displaystyle x^2+4xy+y^2=13 \) at the point \((1,2)\).
- (20 points) Use logarithmic differentiation to find the derivative of
\[y=\left(\left(\tan^{-1}(x)\right)^{\sin^{-1}(x)}\right)\csc^{-1}(x)
\,.\]
- (20 points) Sketch the graph of a single function \(f\) that has all of the
following properties:
- \(f\) is continuous.
- \(\displaystyle \lim_{x\rightarrow -\infty}f(x)=1\).
- \(f\) has a relative minimum at \(x=0\).
- On the interval \([0,2]\), \(f\) has an absolute minimum at \(x=1\).
- On the interval \([0,3]\), \(f\) has an absolute minimum at \(x=3\).
- \(f'(4)=0\)
- \(f'(5) \gt 0\)
- \(f\) has a relative maximum at \(x=6\).
- \(f'(7) \gt 0\)
-
(20 points) For the function \(\displaystyle f(t)=t\sqrt{4-t^2}\),
- find all its critical numbers and
- find its absolute maximum and minimum values on the
interval \([-1,2]\).
-
(20 points) For the function \(\displaystyle f(x)=\ln(x^2+x+1)\),
- find all its critical numbers and
- find its absolute maximum and minimum values on the
interval \([-1,1]\).
Last modified: Thu Feb 17 21:50:37 UTC 2022