Chapter 4

In Problems 25-28, use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\frac{3}{2} \cos x ; x_{1}=1 $$

$$ \int \frac{\left(z^{2}+1\right)^{2}}{\sqrt{z}} d z $$

\(F(x)=6 \sqrt{x}-4 x\) on \([0,4]\)

Find the greatest volume that a right circular cylinder can have if it is inscribed in a sphere of radius \(r\).

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ g(\theta)=\theta^{2} \sec \theta ; I=\left[-\frac{\pi}{4}, \frac{\pi}{4}\right] $$

In Problems 19-28, determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). \(f(x)=\sqrt{\sin x}\) on \([0, \pi]\)

Prove: If \(f\) is continuous on \((a, b)\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x)>0\) except at one point \(x_{0}\) in \((a, b)\), then \(f\) is increasing on \((a, b)\). Hint: Consider \(f\) on each of the intervals \(\left(a, x_{0}\right]\) and \(\left[x_{0}, b\right)\) separately.

The rate of change of volume \(V\) of a melting snowball is proportional to the surface area \(S\) of the ball; that is, \(d V / d t=-k S\), where \(k\) is a positive constant. If the radius of the ball at \(t=0\) is \(r=2\) and at \(t=10\) is \(r=0.5\), show that \(r=-\frac{3}{20} t+2\).

In Problems 25-28, use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=2-\sin x ; x_{1}=2 $$

$$ \int \frac{s(s+1)^{2}}{\sqrt{s}} d s $$

Show that the rectangle with maximum perimeter that can be inscribed in a circle is a square.

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ h(t)=\frac{t^{5 / 3}}{2+t} ; I=[-1,8] $$

In Problems 19-28, determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). \(g(x)=x \sqrt{x-2}\)

From what height must a ball be dropped in order to strike the ground with a velocity of \(-136\) feet per second?

In Problems 25-28, use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{2.7+x} ; x_{1}=1 $$

$$ \int(\sin \theta-\cos \theta) d \theta $$

\(f(x)=\frac{64}{\sin x}+\frac{27}{\cos x}\) on \((0, \pi / 2)\)

What are the dimensions of the right circular cylinder with greatest curved surface area that can be inscribed in a sphere of radius \(r\) ?

Identify the critical points and find the extreme values on the interval \([-1,5]\) for each function: (a) \(f(x)=x^{3}-6 x^{2}+x+2\) (b) \(g(x)=|f(x)|\)

In Problems 19-28, determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). \(f(x)=e^{-x^{2}}\)

Use the Mean Value Theorem to show that \(s=1 / t\) decreases on any interval over which it is defined.

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(0)=0, f(1)=2\); (c) \(f\) is an even function; (d) \(f^{\prime}(x)>0\) for \(x>0\); (e) \(f^{\prime \prime}(x)>0\) for \(x>0\).

In Problems 25-28, use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{3.2+x} ; x_{1}=47 $$

$$ \int\left(t^{2}-2 \cos t\right) d t $$

Inflation between 1999 and 2004 ran at about \(2.5 \%\) per year. On this basis, what would you expect a car that would have cost \( 20,000\) in 1999 to cost in \(2004 ?\)

\(g(x)=x^{2}+\frac{16 x^{2}}{(8-x)^{2}}\) on \((8, \infty)\)

Identify the critical points and find the extreme values on the interval \([-1,5]\) for each function: (a) \(f(x)=\cos x+x \sin x+2\) (b) \(g(x)=|f(x)|\) In Problems 29-36, sketch the graph of a function with the given properties.

Use the Mean Value Theorem to show that \(s=1 / t^{2}\) decreases on any interval to the right of the origin.

If the brakes of a car, when fully applied, produce a constant deceleration of 11 feet per second per second, what is the shortest distance in which the car can be braked to a halt from a speed of 60 miles per hour?

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(2)=-3, f(6)=1\); (c) \(f^{\prime}(2)=0, f^{\prime}(x)>0\) for \(x \neq 2, f^{\prime}(6)=3\); (d) \(f^{\prime \prime}(6)=0, f^{\prime \prime}(x)>0\) for \(26\).

Consider the equation \(x=2\left(x-x^{2}\right)=g(x)\). (a) Sketch the graph of \(y=x\) and \(y=g(x)\) using the same coordinate system, and thereby approximately locate the positive root of \(x=g(x)\). (b) Try solving the equation by the Fixed-Point Algorithm starting with \(x_{1}=0.7\). (c) Solve the equation algebraically.

$$ \int(\sqrt{2} x+1)^{3} \sqrt{2} d x $$

Manhattan Island is said to have been bought by Peter Minuit in 1626 for \( 24\). Suppose that Minuit had instead put the \( 24\) in the bank at \(6 \%\) interest compounded continuously. What would that \(\$ 24\) have been worth in 2000 ?

\(H(x)=\left|x^{2}-1\right|\) on \([-2,2]\)

A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.)

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=1 ; f(6)=3\); increasing and concave down on \((0,6)\)

Prove that if \(F^{\prime}(x)=0\) for all \(x\) in \((a, b)\) then there is a constant \(C\) such that \(F(x)=C\) for all \(x\) in \((a, b)\). Hint: Let \(G(x)=0\) and apply Theorem \(\mathrm{B}\).

What constant acceleration will cause a car to increase its velocity from 45 to 60 miles per hour in 10 seconds?

Sketch the graph of a function \(g\) that has the following properties: (a) \(g\) is everywhere smooth (continuous with a continuous first derivative); (b) \(g(0)=0\) (c) \(g^{\prime}(x)<0\) for all \(x\); (d) \(g^{\prime \prime}(x)<0\) for \(x<0\) and \(g^{\prime \prime}(x)>0\) for \(x>0\).

$$ \int\left(\pi x^{3}+1\right)^{4} 3 \pi x^{2} d x $$

If Methuselah's parents had put \( 100\) in the bank for him at birth and he left it there, what would Methuselah have had at his death ( 969 years later) if interest was \(4 \%\) compounded annually?

\(h(t)=\sin t^{2}\) on \([0, \pi]\)

A closed box in the form of a rectangular parallelepiped with a square base is to have a given volume. If the material used in the bottom costs \(20 \%\) more per square inch than the material in the sides, and the material in the top costs \(50 \%\) more per square inch than that of the sides, find the most economical proportions for the box.

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=8 ; f(6)=-2\); decreasing on \((0,6)\); inflection point at the ordered pair \((2,3)\), concave up on \((2,6)\)

A block slides down an inclined plane with a constant acceleration of 8 feet per second per second. If the inclined plane is 75 feet long and the block reaches the bottom in \(3.75\) seconds, what was the initial velocity of the block?

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(-3)=1\); (c) \(f^{\prime}(x)<0\) for \(x<-3, f^{\prime}(x)>0\) for \(x>-3, f^{\prime \prime}(x)<0\) for \(x \neq-3\).

Consider \(x=\sqrt{1+x}\). (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 \(x=\sqrt{1+x}\). (c) Evaluate \(\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\).

$$ \int\left(5 x^{2}+1\right)\left(5 x^{3}+3 x-8\right)^{6} d x $$

Find the value of \( 1000\) at the end of 1 year when the interest is compounded continuously at \(5 \%\). This is called the future value.

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