§2.3 The product and quotient rules (continued)

Combining rules

Summary

Determine the derivative of the function

. Clearly state which derivative rules you use and how they were applied.

Determine the derivative of the function

. Clearly state which rules you used and how they were applied.

Let $f(r) = (5r^3 + \sin(r))(4^r - 2\cos(r))$. Find $f'(r)$.

Let $\displaystyle p(t) = \frac{\cos(t)}{t^6 \cdot 6^t}$. Find $p'(t)$.

Let $g(z) = 3z^7 e^z - 2z^2 \sin(z) + \frac{z}{z^2 + 1}$. Find $g'(z)$.

A moving particle has its position in feet at time $t$ in seconds given by the function $s(t) = \frac{3\cos(t) - \sin(t)}{e^t}$. Find the particle's instantaneous velocity at the moment $t = 1$.

Suppose that $f(x)$ and $g(x)$ are differentiable functions and it is known that $f(3) = -2$, $f'(3) = 7$, $g(3) = 4$, and $g'(3) = -1$. If $p(x) = f(x) \cdot g(x)$ and $\displaystyle q(x) = \frac{f(x)}{g(x)}$, calculate $p'(3)$ and $q'(3)$.

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 $h(x) = g(x) \cdot f(x)$. Determine $h(2)$ and $h'(2)$. - Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$ (where $h$ is the function defined in (a)). - Let $r$ be the function defined by the rule $r(x) = \frac{g(x)}{f(x)}$. Is $r$ increasing, decreasing, or neither at $a = 2$? Why? - Estimate the value of $r(2.06)$ (where $r$ is the function defined in (c)) by using the local linearization of $r$ at the point $(2,r(2))$.

Let functions $p$ and $q$ be the piecewise linear functions given by their respective graphs in Figure 2.3. Use the graphs to answer the following questions.

• Let $r(x) = p(x) \cdot q(x)$. Determine $r'(-2)$ and $r'(0)$. - Are there values of $x$ for which $r'(x)$ does not exist? If so, which values, and why? - Find an equation for the tangent line to $y = r(x)$ at the point $(2,r(2))$. - Let $z(x) = \frac{q(x)}{p(x)}$. Determine $z'(0)$ and $z'(2)$. - Are there values of $x$ for which $z'(x)$ does not exist? If so, which values, and why?

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 $w(t) = t^t \arccos(t)$. 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 $v(t) = \frac{t^t}{\arccos(t)}$. Is $v$ increasing or decreasing at the instant $t = \frac{1}{2}$? Why?

A farmer with large land holdings has historically grown a wide variety of crops. With the price of ethanol fuel rising, he decides that it would be prudent to devote more and more of his acreage to producing corn. As he grows more and more corn, he learns efficiencies that increase his yield per acre. In the present year, he used 7000 acres of his land to grow corn, and that land had an average yield of 170 bushels per acre. At the current time, he plans to increase his number of acres devoted to growing corn at a rate of 600 acres/year, and he expects that right now his average yield is increasing at a rate of 8 bushels per acre per year. Use this information to answer the following questions. - Say that the present year is $t = 0$, that $A(t)$ denotes the number of acres the farmer devotes to growing corn in year $t$, $Y(t)$ represents the average yield in year $t$ (measured in bushels per acre), and $C(t)$ is the total number of bushels of corn the farmer produces. What is the formula for $C(t)$ in terms of $A(t)$ and $Y(t)$? Why? - What is the value of $C(0)$? What does it measure? - Write an expression for $C'(t)$ in terms of $A(t)$, $A'(t)$, $Y(t)$, and $Y'(t)$. Explain your thinking. - What is the value of $C'(0)$? What does it measure? - Based on the given information and your work above, estimate the value of $C(1)$.

Let $f(v)$ be the gas consumption (in liters/km) of a car going at velocity $v$ (in km/hour). In other words, $f(v)$ tells you how many liters of gas the car uses to go one kilometer if it is traveling at $v$ kilometers per hour. In addition, suppose that $f(80)=0.05$ and $f'(80) = 0.0004$. - Let $g(v)$ be the distance the same car goes on one liter of gas at velocity $v$. What is the relationship between $f(v)$ and $g(v)$? Hence find $g(80)$ and $g'(80)$. - Let $h(v)$ be the gas consumption in liters per hour of a car going at velocity $v$. In other words, $h(v)$ tells you how many liters of gas the car uses in one hour if it is going at velocity $v$. What is the algebraic relationship between $h(v)$ and $f(v)$? Hence find $h(80)$ and $h'(80)$. - How would you explain the practical meaning of these function and derivative values to a driver who knows no calculus? Include units on each of the function and derivative values you discuss in your response.