Chapter 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. $$f(x)=\ln (2 x+x \sin x), \quad[1,15]$$

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

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

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

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

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

Solve the initial value problems in Exercises. $$\frac{d v}{d t}=\frac{3}{t \sqrt{t^{2}-1}}, \quad t>1, v(2)=0$$

Solve the initial value problems in Exercises. $$\frac{d v}{d t}=\frac{8}{1+t^{2}}+\sec ^{2} t, \quad v(0)=1$$

Sketch a smooth connected curve \(y=f(x)\) with \(\begin{aligned}f(-2) &=8 \\\f(0) &=4 \\\f(2) &=0 \\\f^{\prime}(x) &>0 \quad \text { for } \quad|x|>2\end{aligned}\) \(\begin{aligned}&f^{\prime}(2)=f^{\prime}(-2)=0\\\&f^{\prime}(x)<0 \text { for }|x|<2\\\&f^{\prime \prime}(x)<0 \text { for } x<0\\\&f^{\prime \prime}(x)>0 \quad \text { for } \quad x>0\end{aligned}\)

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

Sketch the graph of a twice-differentiable function \(y=f(x)\) with the following properties. Label coordinates where possible. $$\begin{array}{clll}\hline x & y & \ \text { Derivatives } \\\\\hline x<2 & & y^{\prime}<0, & y^{\prime \prime}>0 \\\2 & 1 & y^{\prime}=0, & y^{\prime \prime}>0 \\\2< x<4 & & y^{\prime}>0, & y^{\prime \prime}>0 \\\4 & 4 & y^{\prime}>0, & y^{\prime \prime}=0 \\\4< x<6 & & y^{\prime}>0, & y^{\prime \prime}<0 \\\6 & 7 & y^{\prime}=0, & y^{\prime \prime}<0 \\\x>6 & & y^{\prime}<0, & y^{\prime \prime}<0 \\\\\hline\end{array}$$

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

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

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

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

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

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

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 \(y^{\prime}\)).

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

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?

Solve the initial value problems in Exercises. 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 \(3 \sqrt{x}\)

For \(x>0,\) sketch a curve \(y=f(x)\) that has \(f(1)=0\) and \(f^{\prime}(x)=1 / x .\) Can anything be said about the concavity of such a curve? Give reasons for your answer.

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?

Can anything be said about the graph of a function \(y=f(x)\) that has a continuous second derivative that is never zero? Give reasons for your answer.

If \(b, c,\) and \(d\) are constants, for what value of \(b\) will the curve \(y=x^{3}+b x^{2}+c x+d\) have a point of inflection at \(x=1 ?\) Give reasons for your answer.

Parabolas a. Find the coordinates of the vertex of the parabola \(y=a x^{2}+b x+c, a \neq 0\). b. When is the parabola concave up? Concave down? Give reasons for your answers.

What can you say about the inflection points of a quadratic curve \(y=a x^{2}+b x+c, a \neq 0 ?\) Give reasons for your answer.

What can you say about the inflection points of a cubic curve \(y=a x^{3}+b x^{2}+c x+d, a \neq 0 ?\) Give reasons for your answer.

Finding displacement from an antiderivative of velocity 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 ds/dt 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.

Suppose that the second derivative of the function \(y=f(x)\) is $$y^{\prime \prime}=(x+1)(x-2).$$ For what \(x\) -values does the graph of \(f\) have an inflection point?

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

Suppose that the second derivative of the function \(y=f(x)\) is $$y^{\prime \prime}=x^{2}(x-2)^{3}(x+3).$$ For what \(x\) -values does the graph of \(f\) have an inflection point?

You are driving along a highway at a steady 60 mph \((88 \mathrm{ft} / \mathrm{sec})\) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in \(242 \mathrm{ft}\) ? 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: \(\quad \frac{d s}{d t}=88\) and \(s=0\) when \(t=0\) Measuring time and distance from when the brakes are applied 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 \(s=242\) for the value of \(t\) you found in Step 2

Find the values of constants \(a, b,\) and \(c\) so that the graph of \(y=a x^{3}+b x^{2}+c x\) has a local maximum at \(x=3,\) local minimum at \(x=-1,\) and inflection point at (1,11).

The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph \((44 \mathrm{ft} / \mathrm{sec})\) to 0 in \(45 \mathrm{ft}\). What constant deceleration does it take to do that?

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 \(t\)

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=x^{5}-5 x^{4}-240$$

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 4 \(\mathrm{ft}\) 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 4 ft. How long did it take the hammer and feather to fall 4 ft 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: \(\frac{d^{2} s}{d t^{2}}=-5.2 \mathrm{ft} / \mathrm{sec}^{2}\) Initial conditions: \(\frac{d s}{d t}=0\) and \(s=4\) when \(t=0\)

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=x^{3}-12 x^{2}$$

The standard equation for the position \(s\) of a body moving with a constant acceleration \(a\) along a coordinate line is $$ s=\frac{a}{2} t^{2}+v_{0} t+s_{0} $$ where \(v_{0}\) and \(s_{0}\) are the body's velocity and position at time \(t=0 .\) Derive this equation by solving the initial value problem Differential equation: \(\frac{d^{2} s}{d t^{2}}=a\) Initial conditions: \(\frac{d s}{d t}=v_{0}\) and \(s=s_{0}\) when \(t=0\)

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=\frac{4}{5} x^{5}+16 x^{2}-25$$

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=\frac{x^{4}}{4}-\frac{x^{3}}{3}-4 x^{2}+12 x+20$$

Suppose that $$ f(x)=\frac{d}{d x}(1-\sqrt{x}) \text { and } g(x)=\frac{d}{d x}(x+2) $$ Find: \(\mathbf{a} . \int f(x) d x\) b. \(\int g(x) d x\) c. \(\int[-f(x)] d x\) d. \(\int[-g(x)] d x\) e. \(\int[f(x)+g(x)] d x\) f. \(\int[f(x)-g(x)] d x\)

Graph \(f(x)=2 x^{4}-4 x^{2}+1\) and its first two derivatives together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\) and \(f^{\prime \prime}.\)

If differentiable functions \(y=F(x)\) and \(y=g(x)\) both solve the initial value problem $$ \frac{d y}{d x}=f(x), \quad y\left(x_{0}\right)=y_{0} $$ on an interval \(I,\) must \(F(x)=G(x)\) for every \(x\) in \(I ?\) Give reasons for your answer.

Graph \(f(x)=x \cos x\) and its second derivative together for \(0 \leq x \leq 2 \pi .\) Comment on the behavior of the graph of \(f\) in relation to the signs and values of \(f^{\prime \prime}.\)

Use a CAS to solve the initial value problems in Plot the solution curves.. $$y^{\prime}=\cos ^{2} x+\sin x, \quad y(\pi)=1$$