Chapter 4

A certain rocket, initially at rest, is shot straight up with an acceleration of \(6 t\) meters per second per second during the first 10 seconds after blast- off, after which the engine cuts out and the rocket is subject only to gravitational acceleration of \(-10\) meters per second per second. How high will the rocket go?

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(-4)=-3, f(0)=0, f(3)=2\); (c) \(f^{\prime}(-4)=0, f^{\prime}(3)=0, f^{\prime}(x)>0\) for \(x<-4, f^{\prime}(x)>0\) for \(-43\); (d) \(f^{\prime \prime}(-4)=0, f^{\prime \prime}(0)=0, f^{\prime \prime}(x)<0\) for \(x<-4, f^{\prime \prime}(x)>0\) for \(-40\).

Consider \(x=\sqrt{5+x}\). (a) Apply the Fixed-Point Algorithm starting with \(x_{1}=0\) to find \(x_{2}, x_{3}, x_{4}\), and \(x_{5}\). (b) Algebraically solve for \(x\) in \(x=\sqrt{5+x}\). (c) Evaluate \(\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}\).

$$ \int\left(5 x^{2}+1\right) \sqrt{5 x^{3}+3 x-2} d x $$

Suppose that after 1 year you have \( 1000\) in the bank. If the interest was compounded continuously at \(5 \%\), how much money did you put in the bank one year ago? This is called the present value.

A weight connected to a spring moves along the \(x\)-axis so that its \(x\)-coordinate at time \(t\) is $$ x=\sin 2 t+\sqrt{3} \cos 2 t $$ What is the farthest that the weight gets from the origin?

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=3 ; f(2)=2 ; f(6)=0\); $$ \begin{aligned} &f^{\prime}(x)<0 \text { on }(0,2) \cup(2,6) ; f^{\prime}(2)=0 \\ &f^{\prime \prime}(x)<0 \text { on }(0,1) \cup(2,6) ; f^{\prime \prime}(x)>0 \text { on }(1,2) \end{aligned} $$

Suppose that \(F^{\prime}(x)=5\) and \(F(0)=4\). Find a formula for \(F(x)\). Hint: See Problem 31 .

Starting at station A, a commuter train accelerates at 3 meters per second per second for 8 seconds, then travels at constant speed \(v_{m}\) for 100 seconds, and finally brakes (decelerates) to a stop at station \(B\) at 4 meters per second per second. Find (a) \(v_{m}\) and (b) the distance between \(A\) and \(B\).

Sketch the graph of a function \(f\) that has the following properties. (a) has a continuous first derivative; (b) is decreasing and concave up for \(x<3\); (c) has an extremum at \((3,1)\); (d) is increasing and concave up for \(36\).

Consider \(x=1+\frac{1}{x}\). (a) Apply the Fixed-Point Algorithm starting with \(x_{1}=1\) to find \(x_{2}, x_{3}, x_{4}\), and \(x_{5}\). (b) Algebraically solve for \(x\) in \(x=1+\frac{1}{x}\). (c) Evaluate the following expression. (An expression like this is called a continued fraction.) \(1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}\)

$$ \int 3 t \sqrt[3]{2 t^{2}-11} d t $$

It will be shown later for small \(x\) that \(\ln (1+x) \approx x\). Use this fact to show that the doubling time for money invested at \(p\) percent compounded annually is about \(70 / p\) years.

In Problems 33-38, the first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) local minimum and (b) a local maximum. 33\. \(f^{\prime}(x)=x^{3}(1-x)^{2}\)

A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius \(r\) and vertex angle \(\theta\). Find \(r\) and \(\theta\) if its area is a constant \(A\) and the perimeter is a minimum.

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=f(4)=1 ; f(2)=2 ; f(6)=0\); $$ \begin{aligned} &f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,6) ; \\ &f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime \prime}(x)>0 \text { on }(0,1) \cup(3,4) ; \\ &f^{\prime \prime}(x)<0 \text { on }(1,3) \cup(4,6) \end{aligned} $$

Prove: Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a)\) and \(f(b)\) have opposite signs and if \(f^{\prime}(x) \neq 0\) for all \(x\) in \((a, b)\), then the equation \(f(x)=0\) has one and only one solution between \(a\) and \(b\). Hint: Use the Intermediate Value Theorem and Rolle's Theorem (Problem 22).

Linear approximations provide particularly good approximations near points of inflection. Using a graphing calculator, investigate this behavior in Problems 34-36. Graph \(y=\sin x\) and its linear approximation \(L(x)=x\) at \(x=0\).

Consider the equation \(x=x-f(x) / f^{\prime}(x)\) and suppose that \(f^{\prime}(x) \neq 0\) in an interval \([a, b]\). (a) Show that if \(r\) is in \([a, b]\) then \(r\) is a root of the equation \(x=x-f(x) / f^{\prime}(x)\) if and only if \(f(r)=0\). (b) Show that Newton's Method is a special case of the FixedPoint Algorithm, in which \(g^{\prime}(r)=0\).

$$ \int \frac{3 y}{\sqrt{2 y^{2}+5}} d y $$

The equation for logistic growth is $$ \frac{d y}{d t}=k y(L-y) $$ Show that this differential equation has the solution $$ y=\frac{L y_{0}}{y_{0}+\left(L-y_{0}\right) e^{-L k t}} $$

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=f(3)=3 ; f(2)=4 ; f(4)=2 ; f(6)=0\); $$ \begin{aligned} &f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,5) \\ &f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime}(x)=-1 \text { on }(5,6) \\ &f^{\prime \prime}(x)<0 \text { on }(0,3) \cup(4,5) ; f^{\prime \prime}(x)>0 \text { on }(3,4) \end{aligned} $$

A hot-air balloon left the ground rising at 4 feet per second. Sixteen seconds later, Victoria threw a ball straight up to her friend Colleen in the balloon. At what speed did she throw the ball if it just made it to Colleen?

Experiment with the algorithm $$ x_{n+1}=2 x_{n}-a x_{n}^{2} $$ using several different values of \(a\). (a) Make a conjecture about what this algorithm computes. (b) Prove your conjecture.

$$ \int x^{2} \sqrt{x^{3}+4} d x $$

\(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)(x-4)\)

Prove that a quadratic function has no point of inflection.

\(f\) has domain \([0,6]\), but is not necessarily continuous, and \(f\) does not attain a maximum.

Let \(f\) have a derivative on an interval \(I\). Prove that between successive distinct zeros of \(f^{\prime}\) there can be at most one zero of \(f\). Hint: Try a proof by contradiction and use Rolle's Theorem (Problem 22).

According to Torricelli's Law, the time rate of change of the volume \(V\) of water in a draining tank is proportional to the square root of the water's depth. A cylindrical tank of radius \(10 / \sqrt{\pi}\) centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain. (a) Write the differential equation for \(V\) at time \(t\) and the two corresponding conditions. (b) Solve the differential equation. (c) Find the volume of water after 10 seconds.

$$ \int\left(x^{3}+x\right) \sqrt{x^{4}+2 x^{2}} d x $$

Show that the differential equation $$ \frac{d y}{d t}=a y+b, y(0)=y_{0} $$ has solution $$ y=\left(y_{0}+\frac{b}{a}\right) e^{a t}-\frac{b}{a} $$ Assume that \(a \neq 0\).

\(f^{\prime}(x)=(x-1)^{2}(x-2)^{2}(x-3)^{2}(x-4)^{2}\)

Let \(g\) be continuous on \([a, b]\) and suppose that \(g^{\prime \prime}(x)\) exists for all \(x\) in \((a, b)\). Prove that if there are three values of \(x\) in \([a, b]\) for which \(g(x)=0\) then there is at least one value of \(x\) in \((a, b)\) such that \(g^{\prime \prime}(x)=0\).

The wolf population \(P\) in a certain state has been growing at a rate proportional to the cube root of the population size. The population was estimated at 1000 in 1980 and at 1700 in 1990 . (a) Write the differential equation for \(P\) at time \(t\) with the two corresponding conditions. (b) Solve the differential equation. (c) When will the wolf population reach 4000 ?

$$ \int \sin x(1+\cos x)^{4} d x $$

\(f^{\prime}(x)=(x-A)^{2}(x-B)^{2}, A \neq B\)

Of all right circular cylinders with a given surface area, find the one with the maximum volume. Note: The ends of the cylinders are closed.

Prove that, if \(f^{\prime}(x)\) exists and is continuous on an interval \(I\) and if \(f^{\prime}(x) \neq 0\) at all interior points of \(I\), then either \(f\) is increasing throughout \(I\) or decreasing throughout \(I\). Hint: Use the Intermediate Value Theorem to show that there cannot be two points \(x_{1}\) and \(x_{2}\) of \(I\) where \(f^{\prime}\) has opposite signs.

Suppose \(f^{\prime}(x)=(x-3)(x-2)^{2}(x-1)\) and \(f(2)=0\). Sketch a graph of \(y=f(x)\).

$$ \int \sin x \cos x \sqrt{1+\sin ^{2} x} d x $$

Important news is said to diffuse through an adult population of fixed size \(L\) at a time rate proportional to the number of people who have not heard the news. Five days after a scandal in City Hall was reported, a poll showed that half the people had heard it. How long will it take for \(99 \%\) of the people to hear it?

Find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\).

Prove that if \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\) in \((a, b)\) and if \(x_{1}\) and \(x_{2}\) are any two points in \((a, b)\) then $$ \left|f\left(x_{2}\right)-f\left(x_{1}\right)\right| \leq M\left|x_{2}-x_{1}\right| $$ Note: A function satisfying the above inequality is said to satisfy a Lipschitz condition with constant \(M\). (Rudolph Lipschitz (1832-1903) was a German mathematician.)

Suppose \(h^{\prime}(x)=x^{2}(x-1)^{2}(x-2)\) and \(h(0)=0\). Sketch a graph of \(y=h(x)\).

An object thrown from the edge of a 42 -foot cliff follows the path given by \(y=-\frac{2 x^{2}}{25}+x+42\) (Figure 10 ). An observer stands 3 feet from the bottom of the cliff. (a) Find the position of the object when it is closest to the observer. (b) Find the position of the object when it is farthest from the observer.

$$ \int\left(1+e^{x}\right)^{2} e^{x} d x $$

Show that the relative rate of change of \(e^{k t}\) as a function of \(t\) is \(k\).

In Problems 39-44, sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. 39\. \(f\) is differentiable, has domain \([0,6]\), and has two local maxima and two local minima on \((0,6)\).

Of all rectangles with a given diagonal, find the one with the maximum area.