Chapter 3

Find the derivative of the following functions. $$f(x)=\left(1+\frac{1}{x^{2}}\right)\left(x^{2}+1\right)$$

Use Theorem 3.11 to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\tan 5 x}{x}$$

Find the derivative of the following functions. $$f(v)=v^{100}$$

Equations of tangent lines by definition (1) a. Use definition (1) (p. 127 ) to find the slope of the line tangent to the graph of \(f\) at \(P\) b. Determine an equation of the tangent line at \(P\). c. Plot the graph of \(f\) and the tangent line at \(P\). $$f(x)=5 ; P(1,5)$$

Suppose the position of an object moving horizontally after t seconds is given by the following functions \(s=f(t),\) where \(s\) is measured in feet, with \(s>0\) corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at \(t=1\). d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? $$f(t)=2 t^{2}-9 t+12 ; 0 \leq t \leq 3$$

Balloons A spherical balloon is inflated and its volume increases at a rate of 15 in \(^{3} /\) min. What is the rate of change of its radius when the radius is 10 in?

Evaluate the derivatives of the following functions. $$f(x)=\tan ^{-1} 10 x$$

Find the following derivatives. $$\frac{d}{d x}(\ln |\sin x|)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$\sin x y=x+y$$

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sqrt{x^{2}+1}$$

Find the derivative of the following functions. $$g(w)=e^{w}\left(w^{3}-1\right)$$

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$f(x)=5 x^{3}$$

Equations of tangent lines by definition (1) a. Use definition (1) (p. 127 ) to find the slope of the line tangent to the graph of \(f\) at \(P\) b. Determine an equation of the tangent line at \(P\). c. Plot the graph of \(f\) and the tangent line at \(P\). $$f(x)=\frac{1}{x} ; P(-1,-1)$$

Suppose the position of an object moving horizontally after t seconds is given by the following functions \(s=f(t),\) where \(s\) is measured in feet, with \(s>0\) corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at \(t=1\). d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? $$f(t)=18 t-3 t^{2} ; 0 \leq t \leq 8$$

Evaluate the derivatives of the following functions. $$f(x)=x \cot ^{-1}(x / 3)$$

Find the following derivatives. $$\frac{d}{d x}\left(\frac{\ln x^{2}}{x}\right)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$e^{x y}=2 y$$

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=e^{\sqrt{x}}$$

Find the derivative of the following functions. $$s(t)=4 e^{t} \sqrt{t}$$

Use Theorem 3.11 to evaluate the following limits. $$\lim _{\theta \rightarrow 0} \frac{\sec \theta-1}{\theta}$$

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$g(w)=\frac{5}{6} w^{12}$$

Equations of tangent lines by definition (1) a. Use definition (1) (p. 127 ) to find the slope of the line tangent to the graph of \(f\) at \(P\) b. Determine an equation of the tangent line at \(P\). c. Plot the graph of \(f\) and the tangent line at \(P\). $$f(x)=\frac{4}{x^{2}} ; P(-1,4)$$

Suppose the position of an object moving horizontally after t seconds is given by the following functions \(s=f(t),\) where \(s\) is measured in feet, with \(s>0\) corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at \(t=1\). d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? $$f(t)=2 t^{3}-21 t^{2}+60 t ; 0 \leq t \leq 6$$

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area \(=4 \pi r^{2}\) ).

Evaluate the derivatives of the following functions. $$f(y)=\tan ^{-1}\left(2 y^{2}-4\right)$$

Find the following derivatives. $$\frac{d}{d x}\left(\ln \left(\frac{x+1}{x-1}\right)\right)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$x+y=\cos y$$

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\tan 5 x^{2}$$

a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part \((a)\) $$f(x)=(x-1)(3 x+4)$$

Use Theorem 3.11 to evaluate the following limits. $$\lim _{x \rightarrow 2} \frac{\sin (x-2)}{x^{2}-4}$$

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$p(x)=8 x$$

Equations of tangent lines by definition (2) a. Use definition (2) ( \(p .\) 129) to find the slope of the line tangent to the graph of \(f\) at \(P\). b. Determine an equation of the tangent line at \(P\). $$f(x)=2 x+1 ; P(0,1)$$

Suppose the position of an object moving horizontally after t seconds is given by the following functions \(s=f(t),\) where \(s\) is measured in feet, with \(s>0\) corresponding to positions right of the origin. a. Graph the position function. b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left? c. Determine the velocity and acceleration of the object at \(t=1\). d. Determine the acceleration of the object when its velocity is zero. e. On what intervals is the speed increasing? $$f(t)=-6 t^{3}+36 t^{2}-54 t ; 0 \leq t \leq 4$$

A bug is moving along the right side of the parabola \(y=x^{2}\) at a rate such that its distance from the origin is increasing at \(1 \mathrm{cm} / \mathrm{min} .\) At what rates are the \(x\) - and \(y\) -coordinates of the bug increasing when the bug is at the point (2,4)\(?\)

Evaluate the derivatives of the following functions. $$g(z)=\tan ^{-1}(1 / z)$$

Find the following derivatives. $$\frac{d}{d x}\left(e^{x} \ln x\right)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$x+2 y=\sqrt{y}$$

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sin \frac{x}{4}$$

Use Theorem 3.11 to evaluate the following limits. $$\lim _{x \rightarrow-3} \frac{\sin (x+3)}{x^{2}+8 x+15}$$

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$g(t)=6 \sqrt{t}$$

A bug is moving along the parabola \(y=x^{2} .\) At what point on the parabola are the \(x\) - and \(y\) -coordinates changing at the same rate? (Source: Calculus, Tom M. Apostol, Vol. 1, John Wiley \& Sons, New York, 1967.)

Find the following derivatives. $$\frac{d}{d x}\left(\left(x^{2}+1\right) \ln x\right)$$

Use implicit differentiation to find \(\frac{d y}{d x}\) $$\cos y^{2}+x=e^{y}$$

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sec e^{x}$$

a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part \((a)\) $$g(y)=\left(3 y^{4}-y^{2}\right)\left(y^{2}-4\right)$$

Determine whether the following statements are true and give an explanation or counterexample. a. If the function \(f\) is differentiable for all values of \(x\), then \(f\) is continuous for all values of \(x\). b. The function \(f(x)=|x+1|\) is continuous for all \(x\), but not differentiable for all \(x\). c. It is possible for the domain of \(f\) to be \((a, b)\) and the domain of \(f^{\prime}\) to be \([a, b]\).

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$g(t)=100 t^{2}$$

Equations of tangent lines by definition (2) a. Use definition (2) ( \(p .\) 129) to find the slope of the line tangent to the graph of \(f\) at \(P\). b. Determine an equation of the tangent line at \(P\). $$f(x)=-7 x ; P(-1,7)$$

Suppose a stone is thrown vertically upward from the edge of a cliff on Mars (where the acceleration due to gravity is only about \(12 \mathrm{ft} / \mathrm{s}^{2}\) ) with an initial velocity of \(64 \mathrm{ft} / \mathrm{s}\) from a height of \(192 \mathrm{ft}\) above the ground. The height \(s\) of the stone above the ground after \(t\) seconds is given by \(s=-6 t^{2}+64 t+192\). a. Determine the velocity \(v\) of the stone after \(t\) seconds. b. When does the stone reach its highest point? c. What is the height of the stone at the highest point? d. When does the stone strike the ground? e. With what velocity does the stone strike the ground?

A rectangle initially has dimensions \(2 \mathrm{cm}\) by 4 cm. All sides begin increasing in length at a rate of 1 cm/s. At what rate is the area of the rectangle increasing after \(20 \mathrm{s} ?\)