Overview
1)Using the definition of a derivative, i.e., the limit of the difference quotient, find [Image unavailable: LaTeX: f'\left(a\right)]and the equation of the line tangent to the graph at the given value for a. [Image unavailable: LaTeX: f\left(x\right)=\sqrt[]{3x}], [Image unavailable: LaTeX: a=3].
2)A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
3)Find the derivative of the following functions.
a) [Image unavailable: LaTeX: y=\left(x^2-2x\right)\log_3\left(7x\right)]
b) [Image unavailable: LaTeX: y=\frac{xe^{2x}}{\sin x}]
c) [Image unavailable: LaTeX: y=x^{\sqrt[]{x+1}}]
d) [Image unavailable: LaTeX: y=\sec^{-1}\left(3x^{\frac{1}{2}}\right)]
4)Evaluate the following limit. [Image unavailable: LaTeX: \lim_{x\longrightarrow0}\frac{\sin x}{\sin5x}]
5)Given [Image unavailable: LaTeX: x=\cos\left(x-y\right)], find [Image unavailable: LaTeX: \frac{dy}{dx}]
6)Use the graph of [Image unavailable: LaTeX: g\left(x\right)]to answer the following questions.
[Image unavailable: Graph]
a)Find the values of x in [Image unavailable: LaTeX: \left(-2,2\right)]at which [Image unavailable: LaTeX: g\left(x\right)]is not continuous. Explain why.
b)Find the values of x in [Image unavailable: LaTeX: \left(-2,2\right)]at which [Image unavailable: LaTeX: g\left(x\right)]is not differentiable. Explain why.
7)Find and simplify [Image unavailable: LaTeX: y^{''}]. [Image unavailable: LaTeX: xy+y^2=1]