Chapter 3

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 \(x=a\). $$f(x)=x^{3}-2 x+3, a=2$$

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=2 x+3, \quad a=-1$$

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

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

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

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

Find the first and second derivatives. $$y=-x^{2}+3$$

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

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 \(x=a\). $$f(x)=\sqrt{x^{2}+9}, \quad a=-4$$

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=(1 / 5) x+7, \quad a=-1$$

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

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

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

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. $$y=x^{2}+x+8$$

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

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 \(x=a\). $$f(x)=x+\frac{1}{x}, \quad a=1$$

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=5-4 x, \quad a=1 / 2$$

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

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

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

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

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 \(x=a\). $$f(x)=\sqrt[3]{x}, \quad a=-8$$

a. Find \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) together. c. Evaluate \(d f / d x\) at \(x=a\) and \(d f^{-1} / d x\) at \(x=f(a)\) to show that at these points \(d f^{-1} / d x=1 /(d f / d x)\). $$f(x)=2 x^{2}, \quad x \geq 0, \quad a=5$$

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

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

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

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. $$w=3 z^{7}-7 z^{3}+21 z^{2}$$

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

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 \(x=a\). $$f(x)=\tan x, \quad a=\pi$$

a. Show that \(f(x)=x^{3}\) and \(g(x)=\sqrt[3]{x}\) are inverses of one another. b. Graph \(f\) and \(g\) over an \(x\) -interval large enough to show the graphs intersecting at (1,1) and \((-1,-1) .\) Be sure the picture shows the required symmetry about the line \(y=x\) c. Find the slopes of the tangents to the graphs of \(f\) and \(g\) at (1,1) and (-1,-1) (four tangents in all). d. What lines are tangent to the curves at the origin?

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

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

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

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 the first and second derivatives. $$y=\frac{4 x^{3}}{3}-x+2 e^{x}$$

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

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

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. \(\sin x\) b. \(\cos x\) c. \(\tan x\) d. \(e^{x}\) e. \(\ln (1+x)\)

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

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