Chapter 3

In Exercises \(1-4,\) find the linearization \(L(x)\) of \(f(x)\) at \(x=a\) $$ f(x)=x^{3}-2 x+3, \quad a=2 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=x^{9 / 4} $$

Area Suppose that the radius \(r\) and area \(A=\pi r^{2}\) of a circle are differentiable functions of \(t .\) Write an equation that relates \(d A / d t\) to \(d r / d t .\)

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=6 u-9, \quad u=(1 / 2) x^{4} $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=-10 x+3 \cos x $$

In Exercises 1–12, find the first and second derivatives. $$ y=-x^{2}+3 $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=t^{2}-3 t+2, \quad 0 \leq t \leq 2 $$

Using the definition, calculate the derivatives of the functions in Exercises \(1-6 .\) Then find the values of the derivatives as specified. $$ f(x)=4-x^{2} ; \quad f^{\prime}(-3), f^{\prime}(0), f^{\prime}(1) $$

In Exercises \(1-4,\) find the linearization \(L(x)\) of \(f(x)\) at \(x=a\) $$ f(x)=\sqrt{x^{2}+9}, \quad a=-4 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=x^{-3 / 5} $$

Surface area Suppose that the radius \(r\) and surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t\) . Write an equation that relates \(d S / d t\) to \(d r / d t .\)

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=2 u^{3}, \quad u=8 x-1 $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=\frac{3}{x}+5 \sin x $$

In Exercises 1–12, find the first and second derivatives. $$ y=x^{2}+x+8 $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=6 t-t^{2}, \quad 0 \leq t \leq 6 $$

Using the definition, calculate the derivatives of the functions in Exercises \(1-6 .\) Then find the values of the derivatives as specified. $$ F(x)=(x-1)^{2}+1 ; \quad F^{\prime}(-1), F^{\prime}(0), F^{\prime}(2) $$

In Exercises \(1-4,\) find the linearization \(L(x)\) of \(f(x)\) at \(x=a\) $$ f(x)=x+\frac{1}{x}, \quad a=1 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=\sqrt[3]{2 x} $$

Volume The radius \(r\) and height \(h\) of a right circular cylinder are related to the cylinder's volume \(V\) by the formula \(V=\pi r^{2} h .\) a. How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? b. How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? c. How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=\sin u, \quad u=3 x+1 $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=\csc x-4 \sqrt{x}+7 $$

In Exercises 1–12, find the first and second derivatives. $$ s=5 t^{3}-3 t^{5} $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3 $$

Using the definition, calculate the derivatives of the functions in Exercises \(1-6 .\) Then find the values of the derivatives as specified. $$ g(t)=\frac{1}{t^{2}} ; \quad g^{\prime}(-1), g^{\prime}(2), g^{\prime}(\sqrt{3}) $$

In Exercises \(1-4,\) find the linearization \(L(x)\) of \(f(x)\) at \(x=a\) $$ f(x)=\sqrt[3]{x}, \quad a=-8 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=\sqrt[4]{5 x} $$

Volume The radius \(r\) and height \(h\) of a right circular cone are related to the cone's volume \(V\) by the equation \(V=(1 / 3) \pi r^{2} h .\) a. How is \(d V / d t\) related to \(d h / d t\) if \(r\) is constant? b. How is \(d V / d t\) related to \(d r / d t\) if \(h\) is constant? c. How is \(d V / d t\) related to \(d r / d t\) and \(d h / d t\) if neither \(r\) nor \(h\) is constant?

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=\cos u, \quad u=-x / 3 $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=x^{2} \cot x-\frac{1}{x^{2}} $$

In Exercises 1–12, find the first and second derivatives. $$ w=3 z^{7}-7 z^{3}+21 z^{2} $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=\left(t^{4} / 4\right)-t^{3}+t^{2}, \quad 0 \leq t \leq 3 $$

You want linearizations that will replace the functions in Exercises \(5-10\) over intervals that include the given points \(x_{0} .\) To make your subsequent work as simple as possible, you want to center each linearization not at \(x_{0}\) but at a nearby integer \(x=a\) at which the given function and its derivative are easy to evaluate. What linearization do you use in each case? $$ f(x)=x^{2}+2 x, \quad x_{0}=0.1 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=7 \sqrt{x+6} $$

Changing voltage The voltage \(V\) (volts), current \(I\) (amperes), and resistance \(R\) (ohms) of an electric circuit like the one shown here are related by the equation \(V=I R .\) Suppose that \(V\) is increasing at the rate of 1 volt/sec while \(I\) is decreasing at the rate of 1\(/ 3 \mathrm{amp} / \mathrm{sec}\) . Let \(t\) denote time in seconds. a. What is the value of \(d V / d t ?\) b. What is the value of \(d I / d t ?\) c. What equation relates \(d R / d t\) to \(d V / d t\) and \(d I / d t ?\) d. Find the rate at which \(R\) is changing when \(V=12\) volts and \(I=2\) amp. Is \(R\) increasing, or decreasing?

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=\cos u, \quad u=\sin x $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=(\sec x+\tan x)(\sec x-\tan x) $$

In Exercises 1–12, find the first and second derivatives. $$ y=\frac{4 x^{3}}{3}-x $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=\frac{25}{t^{2}}-\frac{5}{t}, \quad 1 \leq t \leq 5 $$

Find \(d y / d x\) in Exercises \(1-10\) $$ y=-2 \sqrt{x-1} $$

Electrical power The power \(P\) (watts) of an electric circuit is related to the circuit's resistance \(R\) (ohms) and current \(I\) (amperes) by the equation \(P=R I^{2}\) . a. How are \(d P / d t, d R / d t,\) and \(d I / d t\) related if none of \(P, R,\) and \(I\) are constant? b. How is \(d R / d t\) related to \(d I / d t\) if \(P\) is constant?

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=\sin u, \quad u=x-\cos x $$

In Exercises 1–12, find the first and second derivatives. $$ y=\frac{x^{3}}{3}+\frac{x^{2}}{2}+\frac{x}{4} $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=(\sin x+\cos x) \sec x $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$ s=\frac{25}{t+5}, \quad-4 \leq t \leq 0 $$

Distance Let \(x\) and \(y\) be differentiable functions of \(t\) and let \(s=\sqrt{x^{2}+y^{2}}\) be the distance between the points \((x, 0)\) and \((0, y)\) in the \(x y\) -plane. a. How is \(d s / d t\) related to \(d x / d t\) if \(y\) is constant? b. How is \(d s / d t\) related to \(d x / d t\) and \(d y / d t\) if neither \(x\) nor \(y\) is constant? c. How is \(d x / d t\) related to \(d y / d t\) if \(s\) is constant?

Find \(d y / d x\) in Exercises \(1-10\) $$ y=(2 x+5)^{-1 / 2} $$

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$ y=\tan u, \quad u=10 x-5 $$

In Exercises 1–12, find the first and second derivatives. $$ w=3 z^{-2}-\frac{1}{z} $$

In Exercises \(1-12,\) find \(d y / d x\) $$ y=\frac{\cot x}{1+\cot x} $$

Particle motion At time \(t\) , the position of a body moving along the \(s\) -axis is \(s=t^{3}-6 t^{2}+9 t \mathrm{m}\) . a. Find the body's acceleration each time the velocity is zero. b. Find the body's speed each time the acceleration is zero. c. Find the total distance traveled by the body from \(t=0\) to \(t=2\) .