§2.1 Elementary derivative rules (continued)

Summary

Let $f$ and $g$ be differentiable functions for which the following information is known: $f(2) = 5$, $g(2) = -3$, $f'(2) = -1/2$, $g'(2) = 2$. - Let $h$ be the new function defined by the rule $h(x) = 3f(x) - 4g(x)$. Determine $h(2)$ and $h'(2)$. - Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$. - Let $p$ be the function defined by the rule $p(x) = -2f(x) + \frac{1}{2}g(x)$. Is $p$ increasing, decreasing, or neither at $a = 2$? Why? - Estimate the value of $p(2.03)$ by using the local linearization of $p$ at the point $(2,p(2))$.

Let functions $p$ and $q$ be the piecewise linear functions given by their respective graphs in Figure 2.1. Use the graphs to answer the following questions.

• At what values of $x$ is $p$ not differentiable? At what values of $x$ is $q$ not differentiable? Why? - Let $r(x) = p(x) + 2q(x)$. At what values of $x$ is $r$ not differentiable? Why? - Determine $r'(-2)$ and $r'(0)$. - Find an equation for the tangent line to $y = r(x)$ at the point $(2,r(2))$.

Consider the functions $r(t) = t^t$ and $s(t) = \arccos(t)$, for which you are given the facts that $r'(t) = t^t(\ln(t) + 1)$ and $s'(t) = -\frac{1}{\sqrt{1-t^2}}$. Do not be concerned with where these derivative formulas come from. We restrict our interest in both functions to the domain $0 \lt t \lt 1$. - Let $w(t) = 3t^t - 2\arccos(t)$. Determine $w'(t)$. - Find an equation for the tangent line to $y = w(t)$ at the point $(\frac{1}{2}, w(\frac{1}{2}))$. - Let $v(t) = t^t + \arccos(t)$. Is $v$ increasing or decreasing at the instant $t = \frac{1}{2}$? Why?

Let $f(x) = a^x$. The goal of this problem is to explore how the value of $a$ affects the derivative of $f(x)$, without assuming we know the rule for $\frac{d}{dx}[a^x]$ that we have stated and used in earlier work in this section. - Use the limit definition of the derivative to show that

. - Explain why it is also true that

. - Use computing technology and small values of $h$ to estimate the value of

when $a = 2$. Do likewise when $a = 3$. - Note that it would be ideal if the value of the limit $L$ was $1$, for then $f$ would be a particularly special function: its derivative would be simply $a^x$, which would mean that its derivative is itself. By experimenting with different values of $a$ between $2$ and $3$, try to find a value for $a$ for which

. - Compute $\ln(2)$ and $\ln(3)$. What does your work in (b) and (c) suggest is true about $\frac{d}{dx}[2^x]$ and $\frac{d}{dx}[3^x]$? - How do your investigations in (d) lead to a particularly important fact about the function $f(x) = e^x$?