Chapter 4

Determine the values of constants \(a, b, c,\) and \(d\) so that \(f(x)=a x^{3}+b x^{2}+c x+d\) has a local maximum at the point (0,0) and a local minimum at the point (1,-1).

Solve the initial value problems. $$\frac{d y}{d x}=2 x-7, \quad y(2)=0$$

Let \(f\) be differentiable at every value of \(x\) and suppose that \(f(1)=1,\) that \(f^{\prime}<0\) on \((-\infty, 1),\) and that \(f^{\prime}>0\) on \((1, \infty)\) a. Show that \(f(x) \geq 1\) for all \(x\) b. Must \(f^{\prime}(1)=0 ?\) Explain.

If an odd function \(g(x)\) has a local minimum value at \(x=c,\) can anything be said about the value of \(g\) at \(x=-c ?\) Give reasons for your answer.

Solve the initial value problems. $$\frac{d y}{d x}=10-x, \quad y(0)=-1$$

Let \(f(x)=p x^{2}+q x+r\) be a quadratic function defined on a closed interval \([a, b] .\) Show that there is exactly one point \(c\) in \((a, b)\) at which \(f\) satisfies the conclusion of the Mean Value Theorem.

We know how to find the extreme values of a continuous function \(f(x)\) by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{x^{2}}+x, \quad x>0 ; \quad y(2)=1$$

The function $$V(x)=x(10-2 x)(16-2 x), \quad 0 < x < 5$$ models the volume of a box. a. Find the extreme values of \(V\) b. Interpret any values found in part (a) in terms of the volume of the box.

Solve the initial value problems. $$\frac{d y}{d x}=9 x^{2}-4 x+5, \quad y(-1)=0$$

Consider the cubic function $$f(x)=a x^{3}+b x^{2}+c x+d$$ a. Show that \(f\) can have \(0,1,\) or 2 critical points. Give examples and graphs to support your argument. b. How many local extreme values can \(f\) have?

Solve the initial value problems. $$\frac{d y}{d x}=3 x^{-2 / 3}, \quad y(-1)=-5$$

The height of a body moving vertically is given by $$ s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, \quad g > 0 $$ with \(s\) in meters and \(t\) in seconds. Find the body's maximum height.

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{2 \sqrt{x}}, \quad y(4)=0$$

Suppose that at any given time \(t\) (in seconds) the current \(i\) (in amperes) in an alternating current circuit is \(i=2 \cos t+2 \sin t .\) What is the peak current for this circuit (largest magnitude)?

Solve the initial value problems. $$\frac{d s}{d t}=1+\cos t, \quad s(0)=4$$

Graph the functions. Then find the extreme values of the function on the interval and say where they occur. $$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

Solve the initial value problems. $$\frac{d s}{d t}=\cos t+\sin t, \quad s(\pi)=1$$

Graph the functions. Then find the extreme values of the function on the interval and say where they occur. $$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$

Solve the initial value problems. $$\frac{d r}{d \theta}=-\pi \sin \pi \theta, \quad r(0)=0$$

Graph the functions. Then find the extreme values of the function on the interval and say where they occur. $$h(x)=|x+2|-|x-3|, \quad-\infty < x < \infty$$

Solve the initial value problems. $$\frac{d r}{d \theta}=\cos \pi \theta, \quad r(0)=1$$

Graph the functions. Then find the extreme values of the function on the interval and say where they occur. $$k(x)=|x+1|+|x-3|, \quad-\infty < x < \infty$$

Solve the initial value problems. $$\frac{d v}{d t}=\frac{1}{2} \sec t \tan t, \quad v(0)=1$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]$$

Solve the initial value problems. $$\frac{d v}{d t}=8 t+\csc ^{2} t, \quad v\left(\frac{\pi}{2}\right)=-7$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]$$

Solve the initial value problems. $$\frac{d^{2} y}{d x^{2}}=2-6 x ; \quad y^{\prime}(0)=4, \quad y(0)=1$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{2 / 3}(3-x), \quad[-2,2]$$

Solve the initial value problems. $$\frac{d^{2} y}{d x^{2}}=0 ; \quad y^{\prime}(0)=2, \quad y(0)=0$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]$$

Solve the initial value problems. $$\frac{d^{2} r}{d t^{2}}=\frac{2}{t^{3}} ;\left.\quad \frac{d r}{d t}\right|_{t=1}=1, \quad r(1)=1$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]$$

Solve the initial value problems. $$\frac{d^{2} s}{d t^{2}}=\frac{3 t}{8} ;\left.\quad \frac{d s}{d t}\right|_{t=4}=3, \quad s(4)=4$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$$

Solve the initial value problems. $$\frac{d^{3} y}{d x^{3}}=6 ; \quad y^{\prime \prime}(0)=-8, \quad y^{\prime}(0)=0, \quad y(0)=5$$

Solve the initial value problems. $$\frac{d^{3} \theta}{d t^{3}}=0 ; \quad \theta^{\prime \prime}(0)=-2, \quad \theta^{\prime}(0)=-\frac{1}{2}, \quad \theta(0)=\sqrt{2}$$

Solve the initial value problems. $$\begin{aligned} &y^{(4)}=-\sin t+\cos t\\\ &y^{\prime \prime \prime}(0)=7, \quad y^{\prime \prime}(0)=y^{\prime}(0)=-1, \quad y(0)=0 \end{aligned}$$

Solve the initial value problems. $$\begin{array}{l} y^{(4)}=-\cos x+8 \sin 2 x \\ y^{\prime \prime \prime}(0)=0, \quad y^{\prime \prime}(0)=y^{\prime}(0)=1, \quad y(0)=3 \end{array}$$

Find the curve \(y=f(x)\) in the \(x y\) -plane that passes through the point (9,4) and whose slope at each point is \(3 \sqrt{x}\).

a. Find a curve \(y=f(x)\) with the following properties: i) \(\frac{d^{2} y}{d x^{2}}=6 x\) ii) Its graph passes through the point (0,1) and has a horizontal tangent there. b. How many curves like this are there? How do you know?

Sketch a smooth connected curve \(y=f(x)\) with \(f(-2)=8, \quad f^{\prime}(2)=f^{\prime}(-2)=0\), \(f(0)=4, \quad f^{\prime}(x)<0 \quad\) for \(\quad|x|<2\), \(f(2)=0, \quad f^{\prime \prime}(x)<0 \quad\) for \(\quad x<0\), \(f^{\prime}(x)>0\) for \(|x|>2, \quad f^{\prime \prime}(x)>0\) for \(x>0\).

a. Suppose that the velocity of a body moving along the \(s\) -axis is $$\frac{d s}{d t}=v=9.8 t-3$$ i) Find the body's displacement over the time interval from \(t=1\) to \(t=3\) given that \(s=5\) when \(t=0\) ii) Find the body's displacement from \(t=1\) to \(t=3\) given that \(s=-2\) when \(t=0\) iii) Now find the body's displacement from \(t=1\) to \(t=3\) given that \(s=s_{0}\) when \(t=0\) b. Suppose that the position \(s\) of a body moving along a coordinate line is a differentiable function of time \(t\). Is it true that once you know an antiderivative of the velocity function \(d s / d t\) you can find the body's displacement from \(t=a\) to \(t=b\) even if you do not know the body's exact position at either of those times? Give reasons for your answer.

Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of \(20 \mathrm{m} / \mathrm{s}^{2} .\) How fast will the rocket be going 1 min later?

Stopping a car in time You are driving along a highway at a steady \(108 \mathrm{km} / \mathrm{h}(30 \mathrm{m} / \mathrm{s})\) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in \(75 \mathrm{m} ?\) To find out, carry out the following steps. 1. Solve the initial value problem Differential equation: \(\frac{d^{2} s}{d t^{2}}=-k\) \((k \text { constant })\) Initial conditions: \(\quad \frac{d s}{d t}=30\) and \(s=0\) when \(t=0\) 2\. Find the value of \(t\) that makes \(d s / d t=0 .\) (The answer will involve \(k .)\) 3\. Find the value of \(k\) that makes \(s=75\) for the value of \(t\) you found in Step 2.

Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from \(48 \mathrm{km} / \mathrm{h}(13.3 \mathrm{m} / \mathrm{s})\) to 0 in \(13.7 \mathrm{m} .\) What constant deceleration does it take to do that?

Motion along a coordinate line A particle moves on a coordinate line with acceleration \(a=d^{2} s / d t^{2}=15 \sqrt{t}-(3 / \sqrt{t})\) subject to the conditions that \(d s / d t=4\) and \(s=0\) when \(t=1\) Find a. the velocity \(v=d s / d t\) in terms of \(t\) b. the position \(s\) in terms of \(t\).

Suppose the derivative of the function \(y=f(x)\) is \(y^{\prime}=(x-1)^{2}(x-2)\). At what points, if any, does the graph of \(f\) have a local minimum, local maximum, or point of inflection? (Hint: Draw the sign pattern for \(y^{\prime} .\) )

The hammer and the feather When Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon to demonstrate that in a vacuum all bodies fall with the same (constant) acceleration, he dropped them from about \(1.2 \mathrm{m}\) above the ground. The television footage of the event shows the hammer and the feather falling more slowly than on Earth, where, in a vacuum, they would have taken only half a second to fall the \(1.2 \mathrm{m}\). How long did it take the hammer and feather to fall \(1.2 \mathrm{m}\) on the moon? To find out, solve the following initial value problem for \(s\) as a function of \(t .\) Then find the value of \(t\) that makes \(s\) equal to \(0 .\). Differential equation: \(\frac{d^{2} s}{d t^{2}}=-1.6 \mathrm{m} / \mathrm{s}^{2}\) Initial conditions: \(\frac{d s}{d t}=0\) and \(s=1.2\) when \(t=0\)

Suppose the derivative of the function \(y=f(x)\) is $$y^{\prime}=(x-1)^{2}(x-2)(x-4)$$. At what points, if any, does the graph of \(f\) have a local minimum, local maximum, or point of inflection?