Chapter 3

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

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 \(d r / d t .\)

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=x^{3}-2 x+3, \quad a=2\)

Use implicit differentiation to find \(d y / d x\). \(x^{2} y+x y^{2}=6\)

Find \(d y / d x\) $$ y=-10 x+3 \cos x $$

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? $$s=t^{2}-3 t+2, \quad 0 \leq t \leq 2$$

In Exercises \(1-12,\) find the first and second derivatives. \begin{equation} y=-x^{2}+3 \end{equation}

Using the definition, calculate the derivatives of the functions. 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 \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(d y / d x=f^{\prime}(g(x)) g^{\prime}(x)\) $$ y=2 u^{3}, \quad u=8 x-1 $$

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 \(d r / d t\)

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=\sqrt{x^{2}+9}, \quad a=-4\)

Use implicit differentiation to find \(d y / d x\). \(x^{3}+y^{3}=18 x y\)

Find \(d y / d x\) $$ y=\frac{3}{x}+5 \sin x $$

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? $$s=6 t-t^{2}, \quad 0 \leq t \leq 6$$

Find the first and second derivatives. \begin{equation} y=x^{2}+x+8 \end{equation}

Using the definition, calculate the derivatives of the functions. 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 \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(d y / d x=f^{\prime}(g(x)) g^{\prime}(x)\) $$ y=\sin u, \quad u=3 x+1 $$

Assume that \(y=5 x\) and \(d x / d t=2 .\) Find \(d y / d t\)

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=x+\frac{1}{x}, \quad a=1\)

Use implicit differentiation to find \(d y / d x\). \(2 x y+y^{2}=x+y\)

Find \(d y / d x\) $$ y=x^{2} \cos x $$

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? $$s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3$$

Find the first and second derivatives. $$ s=5 t^{3}-3 t^{5} $$

Using the definition, calculate the derivatives of the functions. 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 \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(d y / d x=f^{\prime}(g(x)) g^{\prime}(x)\) $$ y=\cos u, u=-\frac{x}{3} $$

Assume that \(2 x+3 y=12\) and \(d y / d t=-2 .\) Find \(d x / d t\)

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=\sqrt[3]{x}, \quad a=-8\)

Use implicit differentiation to find \(d y / d x\). \(x^{3}-x y+y^{3}=1\)

Find \(d y / d x\) $$ y=\sqrt{x} \sec x+3 $$

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? $$s=\left(t^{4} / 4\right)-t^{3}+t^{2}, \quad 0 \leq t \leq 3$$

Find the first and second derivatives. $$ w=3 z^{7}-7 z^{3}+21 z^{2} $$

derivatives as specified. $$ k(z)=\frac{1-z}{2 z} ; \quad k^{\prime}(-1), k^{\prime}(1), k^{\prime}(\sqrt{2}) $$

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

If \(y=x^{2}\) and \(d x / d t=3,\) then what is \(d y / d t\) when \(x=-1 ?\)

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=\tan x, \quad a=\pi\)

Use implicit differentiation to find \(d y / d x\). \(x^{2}(x-y)^{2}=x^{2}-y^{2}\)

Find \(d y / d x\) $$ y=\csc x-4 \sqrt{x}+7 $$

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 an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together. \(y=4-x^{2}, \quad(-1,3)\)

Find the first and second derivatives. \begin{equation} y=\frac{4 x^{3}}{3}-x \end{equation}

Using the definition, calculate the derivatives of the functions. Then find the values of the derivatives as specified. $$ p(\theta)=\sqrt{3 \theta} ; \quad p^{\prime}(1), p^{\prime}(3), p^{\prime}(2 / 3) $$

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

If \(x=y^{3}-y\) and \(d y / d t=5,\) then what is \(d x / d t\) when \(y=2 ?\)

Common linear approximations at \(x=0\) Find the linearizations of the following functions at \(x=0.\) a. \(\sin x \quad\) b. \(\cos x \quad\) c. \(\tan x\)

Use implicit differentiation to find \(d y / d x\). \((3 x y+7)^{2}=6 y\)

Find \(d y / d x\) $$ y=x^{2} \cot x-\frac{1}{x^{2}} $$

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? $$s=\frac{25}{t+5}, \quad-4 \leq t \leq 0$$

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together. \(y=(x-1)^{2}+1, \quad(1,1)\)

Find the first and second derivatives. \begin{equation} y=\frac{x^{3}}{3}+\frac{x^{2}}{2}+\frac{x}{4} \end{equation}

Using the definition, calculate the derivatives of the functions. Then find the values of the derivatives as specified. $$ r(s)=\sqrt{2 s+1} ; \quad r^{\prime}(0), r^{\prime}(1), r^{\prime}(1 / 2) $$