Lectures>Explorations>Practice

# Exercises

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## Exercises

Use the given information to find the equation of the tangent line to the function at the given point.

 $y=f(x)$ at $x=2$ if $f(2) = 5$ and $f'(2) = -1$ $y=f(x)$ at $x=-1$ if $f(-1) = 2$ and $f'(-1) = 0$ $f(x) = x^2 + 1$ at $x=2$ if $f'(2) = 4$. $f(x) = \cos(x)$ at $x=\pi/2$ if $f'(\pi/2) = -1$.

Find the equation of the tangent line to the following equations at the given point. Graph the function and the tangent line at that point.

 $f(x)=x^2+2$ at (1, 3) $f(x)=4x-3$ at (0, 3) $f(x)=\frac{1}{x}$ at (1, 1) $f(x)=\sqrt{x+1}$ at (3, 2)

Each of the following limits represents the derivative of a function, f, at a point, $x=a$. Determine both the function and the point.

 $\lim\limits_{h\to 0} \frac{\sqrt{h+4}-2}{h}$ $\lim\limits_{h\to 0} \frac{\ln{(h+1)}}{h}$ $\lim\limits_{h\to 0} \frac{\cos{h}-1}{h}$ $\lim\limits_{h\to 0} \frac{(x-2)^2 - 4}{h}$

For each function below, find the slope of the tangent line at $x = a$.

 $f(x) = x^2 - 4x + 2$ $f(x) = \frac{x+1}{x}$ $f(x) = \frac{2}{x^2}$ $f(x) = (x+1)^{-2}$

Using the information below, slope of the tangent line and the value of the function at the given point.

1. If the equation of the tangent line to y at $x=2$ is $y=3x-1$, what are $f(2)$ and $f'(2)$.
2. If the equation of the tangent line to y at $x=-1$ is $y=-x+4$, what are $f(-1)$ and $f'(-1)$.

Sketch the graph of a function that meets the following requirements.

1. $f(1) = 0$, $f'(0) = 0$ and f is decreasing to the left of 0.
2. $f(0) = 0$, $f'(1) = f'(-1) = 1$.
3. $f(\pm 2) = 0$, $f'(0) = 0$, $\lim_\limits{x\to\infty}f(x) = 1$.
4. $f(2) = 3$, $f'(-1) = 0$, $f'(1)=2$, $\lim_\limits{x\to-\infty}f(x) = 0$ and $\lim_\limits{x\to\infty}f(x) = \infty$.

Determine which of the following functions are not differentiable at the origin.

 $g(x) = \begin{cases} x^2+1, & x \le 0 \\ 0, & x \gt 0 \end{cases}$ $g(x) = \begin{cases} 3x+1, & x \le 0 \\ -2x+1, & x \gt 0 \end{cases}$ $g(x) = \begin{cases} 3x-1, & x \le 0 \\ 3x+1, & x \gt 0 \end{cases}$ $f(x) = \sqrt[3]{x}$. $f(x) = \sqrt[3]{x^2}$. $f(x) = \sqrt[4]{x}$.

## Explorations

For each of the functions below, find the instantaneous rate of change at the given point. Graph the function and the tangent lines at those points and think about the relationship between the sign of the tangent line's slope and the behavior of the function at each point.

 $f(x) = x^2$ at $x=-1, 0, 1$. $f(x) = -x^2 + 4x - 4$ at $x=1, 2, 3$ $f(x) = x^3$ at $x=-1, 0, 1$

## Applications

1. Newton's Law of Gravitation says that the force between two objects whose masses are $m_1$ and $m_2$ and that are r meters apart is given by $F=G\frac{m_1m_2}{r^2}$ where G is Newton's gravitation constant. Assuming the two masses stay constant, how fast is the force changing when the two objects are 1 million meters apart. How fast is it changing when the distance between the objects is 1 meter? Use $G=6.67430 \times 10^{-37}$.
2. The demand, Q, for a product in thousands for a given price, p, is given by $Q(p) = -x^2-6x+40$. It may seem unusual for the demand to increase, at least initially, as the price increases but this is a known psychological phenomenon in marketing. A more expensive product is perceived as being higher quality which can increase demand up to a certain point at which the perceived value no longer matches the asked price and the demand will start to decrease. Determine the rate of change of the demand at $x=a$. Based on your result, at what price do you think the demand stops increasing and starts to decrease.

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