Give an example of:

Two different functions that have the same linear approximation near $x = 0$.

Assume $a, b, c,$ and $k$ are constants.

$$f(s) = \frac{a^2 − s^2}{\sqrt{a^2 + s^2}}$$

Find the derivatives of the functions. Assume $a$, $b$, and $c$ are constants.

$$y = a \sin(bt) + c$$

True or false? Explain your answer: If $y$ satisfies the equation $y^2 + xy − 1=0$, then $dy/dx$ exists everywhere.

Find the derivative of the functions. $f(\theta)=\left(e^{\theta}+e^{-\theta}\right)^{-1}$

To compare the acidity of different solutions, chemists use the $pH$ (which is a single number, not the product of $p$ and $H$). The $pH$ is defined in terms of the concentration, $x$, of hydrogen ions in the solution as $pH = − \log x$. Find the rate of change of $pH$ with respect to hydrogen ion concentration when the $pH$ is $2$. [Hint: Use the result of Problem $45$.]

Find the equation of the tangent line at $x = 1$ to $y = f(x)$ where $f(x) = \frac{3x^2} {5x^2 + 7x}$

Are the statements true or false? Give an explanation for your answer.

The function $\sinh2 x$ is concave down everywhere.

Find the derivatives. Assume that $a$ and $b$ are constants.

$$f(w) = (5w^2 + 3)e^{w^2}$$

A curve that has two horizontal tangents at the same $x$-value, but no vertical tangents.

Find the equation of the tangent line to the graph of $f(x)=\frac{2 x-5}{x+1}$ at the point at which x=0.

Find the quadratic polynomial $g(x)=a x^{2}+b x+c$ which best fits the function $f(x)=e^{x}$ at x=0, in the sense that g(0)=f(0), and g'(0)=f'(0), and g''(0)=f''(0). Using a computer or calculator, sketch graphs off and g on the same axes. What do you notice?

Find the derivative. Assume a, b, c, k are constants. $h(x)=\frac{a x+b}{c}$

Explain what is wrong with the statement.

To approximate $f(x) = e^x$, we can always use the linear approximation $f(x) = e^x \approx x + 1$.