Use the definition of the derivative to compute the derivative of \(f\text{.}\)
1.
\(f(x)=5x^2\)
2.
\(f(x)=(x-2)^3\)
Exercise Group.
Numerically approximate the derivative.
3.
\(f'(\pi)\) where \(f(x) = \cos(x)\)
4.
\(f'(16)\) where \(f(x) = \sqrt{x}\)
5.
Given that \(e^0=1\text{,}\) approximate the value of \(e^{0.1}\) using the tangent line to \(f(x) = e^x\) at \(x=0\text{.}\)
6.
Approximate the value of \((3.1)^{4}\) using the tangent line to \(f(x) = {x^{4}}\) at \(x=3\text{.}\)
Exercise Group.
Use the graph of \(f(x)\) to sketch \(\fp(x)\text{.}\)
7.
8.
9.
10.
11.
The “wind chill factor” is a measurement of how cold it “feels” during cold, windy weather. Let \(W(w)\) be the wind chill factor, in degrees Fahrenheit, when it is \(25^\circ\)F outside with a wind of \(w\) mph.
(a)
What are the units of \(W'(w)\text{?}\)
(b)
What would you expect the sign of \(W'(10)\) to be?
Positive
Negative
12.
Find the derivatives of the following functions.
(a)
\(f(x) = {5xe^{x}\cot\mathopen{}\left(x\right)}\)
(b)
\(g(x) = {4^{x}\cdot 7^{x}\cdot 9^{x}}\)
13.
Find \(\lz{y}{x}\text{,}\) where \({x^{2}y^{3}-y^{4}x}=1\text{.}\)
14.
Find the equation of the line tangent to the graph of \({x^{4}+y^{4}+xy}={15}\) at the point \((-1,2)\text{.}\)
15.
Let \(f(x) = {x^{4}-2x}\text{.}\) Find \(\lim\limits_{s\to 0} \frac{f(x+s)-f(x)}{s}\text{.}\)
16.
Find \(\lz{y}{x}\text{,}\) where \({x^{2}y^{4}-y^{3}x}=1\text{.}\)
17.
Find the equation of the line tangent to the graph of \({x^{2}+y^{2}+xy}={7}\) at the point \((-1,3)\text{.}\)
18.
Let \(f(x) = {x^{3}-2x}\text{.}\) Find \(\lim\limits_{s\to 0} \frac{f(x+s)-f(x)}{s}\text{.}\)
