Chapter 2

Moving Point A point is moving along the graph of \(y=1 /\left(1+x^{2}\right)\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x .\) $$ \text { (a) } x=-2 \quad \text { (b) } x=2 \quad \text { (c) } x=0 \quad \text { (d) } x=10 $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{3}-y^{2}=0\) \(\quad\) \((1,1)\)

find the second derivative of the function. $$ h(s)=s^{3}\left(s^{2}-2 s+1\right) $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 x+4 ;(1,6) $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\\{g(x)=\frac{4 x-5}{x^{2}-1}} & {(0,5)} \end{array}$$

Velocity The height \(s(\text { in feet) at time } t\) (in seconds) of a silver dollar dropped from the top of the Washington Monument is given by $$ s=-16 t^{2}+555 $$ (a) Find the average velocity on the interval \([2,3] .\) (b) Find the instantaneous velocities when \(t=2\) and when \(t=3 .\) (c) How long will it take the dollar to hit the ground? (d) Find the velocity of the dollar when it hits the ground.

Find the derivative of the function. $$ y=4 t^{4 / 3} $$

Moving Ladder A 25 -foot ladder is leaning against a house (see figure). The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base is (a) 7 feet, \((b) 15\) feet, and (c) 24 feet from the house?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{3}-x y+y^{2}=4 \quad(0,-2)\)

find the third derivative of the function. $$ f(x)=x^{5}-3 x^{4} $$

Physics: Velocity A racecar travels northward on a straight, level track at a constant speed, traveling 0.750 kilometer in 20.0 seconds. The return trip over the same track is made in 25.0 seconds. (a) What is the average velocity of the car in meters per second for the first leg of the run? (b) What is the average velocity for the total trip?

Find the derivative of the function. $$ h(x)=x^{5 / 2} $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{2} y+y^{2} x=-2\) \(\quad\) \((2,-1)\)

find the third derivative of the function. $$ f(x)=x^{4}-2 x^{3} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6 ;(-2,6) $$

Find the derivative of the function. $$ f(x)=4 \sqrt{x} $$

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 450 miles per hour. The other is 200 miles from the point and has a speed of 600 miles per hour. (a) At what rate is the distance between the planes changing? (b) How much time does the controller have to get one of the airplanes on a different flight path?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{3} y^{3}-y=x\) \(\quad\) \((0,0)\)

find the third derivative of the function. $$ f(x)=5 x(x+4)^{3} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{2}-1 ;(2,3) $$

Find the marginal cost for producing units. (The cost is measured in dollars.) $$ C=205,000+9800 x $$

Find the derivative of the function. $$ g(x)=4 \sqrt[3]{x}+2 $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{3}+y^{3}=2 x y\) \(\quad\) \((1,1)\)

find the third derivative of the function. $$ f(x)=\left(x^{3}-6\right)^{4} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=4-x^{2} ;(2,0) $$

Find the marginal cost for producing units. (The cost is measured in dollars.) $$ C=55,000+470 x-0.25 x^{2}, \quad 0 \leq x \leq 940 $$

Find the derivative of the function. $$ y=4 x^{-2}+2 x^{2} $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{1 / 2}+y^{1 / 2}=9\) \(\quad\) \((16,25)\)

find the third derivative of the function. $$ f(x)=\frac{3}{16 x^{2}} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{3}-x ;(2,6) $$

Find the marginal cost for producing units. (The cost is measured in dollars.) $$ C=100(9+3 \sqrt{x}) $$

Find the derivative of the function. $$ s(t)=4 t^{-1}+1 $$

Advertising costs A retail sporting goods store estimates that weekly sales \(S\) and weekly advertising costs \(x\) are related by the equation \(S=2250+50 x+0.35 x^{2} .\) The current weekly advertising costs are \(\$ 1500,\) and these costs are increasing at a rate of \(\$ 125\) per week. Find the current rate of change of weekly sales.

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(\sqrt{x y}=x-2 y\) \(\quad\) \((4,1)\)

Match the function with the rule that you would use to find the derivative most efficiently. \(\begin{array}{ll}{\text { (a) Simple Power Rule }} & {\text { (b) Constant Rule }} \\ {\text { (c) General Power Rule }} & {\text { (d) Quotient Rule }}\end{array}\) $$ f(x)=\frac{5}{x^{2}+1} $$

find the third derivative of the function. $$ f(x)=\frac{1}{x} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{3}+2 x ;(1,3) $$

Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=50 x-0.5 x^{2} $$

An accident at an oil drilling platform is causing a circular oil slick. The slick is 0.08 foot thick, and when the radius of the slick is 150 feet, the radius is increasing at the rate of 0.5 foot per minute. At what rate (in cubic feet per minute) is oil flowing from the site of the accident?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{2 / 3}+y^{2 / 3}=5\) \(\quad\) \((8,1)\)

Use the General Power Rule to find the derivative of the function. $$ y=(2 x-7)^{3} $$

find the given value. $$ g(t)=5 t^{4}+10 t^{2}+3 \quad g^{\prime \prime}(2) $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 \sqrt{x} ;(4,4) $$

Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=30 x-x^{2} $$

Use Example 6 as a model to find the derivative. $$ y=\frac{2}{3 x^{2}} $$

Profit A company is increasing the production of a product at the rate of 25 units per week. The demand and cost functions for the product are given by \(p=50-0.01 x\) and \(C=4000+40 x-0.02 x^{2} .\) Find the rate of change of the profit with respect to time when the weekly sales are \(x=800\) units. Use a graphing utility to graph the profit function, and use the zoom and trace features of the graphing utility to verify your result.

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \((x+y)^{3}=x^{3}+y^{3} \quad(-1,1)\)

Use the General Power Rule to find the derivative of the function. $$ y=\left(2 x^{3}+1\right)^{2} $$

find the given value. $$ f(x)=9-x^{2} \quad f^{\prime \prime}(-\sqrt{5}) $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=\sqrt{x+1} ;(8,3) $$