Chapter 6

Evaluate the following derivatives. $$\frac{d}{d x}\left(\ln ^{3}\left(3 x^{2}+2\right)\right)$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\sin x ;[0, \pi]$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}} \text { on }[1,2]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=\sqrt{x}, y=0, \text { and } x=4$$

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by \(y=x^{2}\) and the line \(y=1\) and whose cross sections perpendicular to the base and parallel to the \(x\) -axis are squares

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=50 e^{-2 t} ; 0 \leq t \leq 4$$

Verify each identity using the definitions of the hyperbolic functions. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) (Hint: Begin with the right side of the equation.)

Evaluate the following integrals. Include absolute values only when needed. $$\int_{0}^{3} \frac{2 x-1}{x+1} d x$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\ln x ;[1,4]$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\frac{x^{3}}{3}+\frac{1}{4 x} \text { on }\left[\frac{1}{2}, 2\right]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=\cos x^{2}, y=0, \text { for } 0 \leq x \leq \sqrt{\pi / 2}$$

Use the general slicing method to find the volume of the following solids. The solid whose base is the triangle with vertices (0,0),(2,0) and (0,2) and whose cross sections perpendicular to the base and parallel to the \(y\) -axis are semicircles

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The current population of a town is 50,000 and is growing exponentially. If the population is projected to be 55,000 in 10 years, then what will be the population 20 years from now?

Sketch the region and find its area. The region bounded by \(y=\frac{2}{1+x^{2}}\) and \(y=1\)

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. How long will it take an initial deposit of \(\$ 1500\) to increase in value to \(\$ 2500\) in a saving account with an APY of 3.1\%? Assume the interest rate remains constant and no additional deposits or withdrawals are made.

Evaluate the following integrals. Include absolute values only when needed. $$\int \tan 10 x d x$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\frac{x^{3}}{3} ;[-1,1]$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\sqrt{5 x-x^{2}} \text { on }[1,4]$$

The pyramid with a square base \(4 \mathrm{m}\) on a side and a height of \(2 \mathrm{m}\) (Use calculus.)

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the anti- derivative method and the Fundamental Theorem of Calculus (Theorem 1). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=-t^{3}+3 t^{2}-2 t \text { on }[0,3] ; s(0)=4$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. \(y=\sqrt{4-2 x^{2}}, y=0,\) and \(x=0,\) in the first quadrant

Sketch the region and find its area. The region bounded by \(y=24 \sqrt{x}\) and \(y=3 x^{2}\)

Find the mass of the following thin bars with the given density function. $$\rho(x)=\left\\{\begin{array}{ll} 1 & \text { if } 0 \leq x \leq 2 \\ 1+x & \text { if } 2

Verify each identity using the definitions of the hyperbolic functions. \(\cosh x+\sinh x=e^{x}\)

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. Between 2005 and \(2010,\) the average rate of inflation was about \(3 \% /\) yr (as measured by the Consumer Price Index). If a cart of groceries cost \(\$ 100\) in \(2005,\) what will it cost in 2015 assuming the rate of inflation remains constant?

Evaluate the following integrals. Include absolute values only when needed. $$\int_{e}^{e^{2}} \frac{d x}{x \ln ^{3} x}$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\sqrt{x-2} ;[3,4]$$

\(A\) 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that \(x\) and \(y\) are measured in meters. The spherical zone generated when the curve \(y=\sqrt{8 x-x^{2}}\) on the interval [1,7] is revolved about the \(x\) -axis

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=\sqrt{x}, y=0, \text { and } x=4$$

The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4

Sketch the following regions (if a figure is not given) and then find the area. The region bounded by \(y=\sin x, y=\cos x,\) and the \(x\) -axis between \(x=0\) and \(x=\pi / 2\)

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the anti- derivative method and the Fundamental Theorem of Calculus (Theorem 1). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=6-2 t \text { on }[0,5] ; s(0)=0$$

Find the mass of the following thin bars with the given density function. $$\rho(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } 0 \leq x \leq 1 \\ x(2-x) & \text { if } 1

Use the given identity to verify the related identity. Use the fundamental identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to verify the identity \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x\).

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?

Evaluate the following integrals. Include absolute values only when needed. $$\int_{0}^{\pi / 2} \frac{\sin x}{1+\cos x} d x$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\frac{8}{x^{2}} ;[1,4]$$

\(A\) 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that \(x\) and \(y\) are measured in meters. The spherical zone generated when the upper portion of the circle \(x^{2}+y^{2}=100\) on the interval [-8,8] is revolved about the \(x\) -axis

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=8, y=2 x+2, x=0, \text { and } x=2$$

Sketch the following regions (if a figure is not given) and then find the area. The regions between \(y=\sin x\) and \(y=\sin 2 x,\) for \(0 \leq x \leq \pi\)

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the anti- derivative method and the Fundamental Theorem of Calculus (Theorem 1). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=3 \sin \pi t \text { on }[0,4] ; s(0)=1$$

How much work is required to move an object from \(x=0\) to \(x=3\) (measured in meters) in the presence of a force (in \(\mathrm{N}\) ) given by \(F(x)=2 x\) acting along the \(x\) -axis?

Use the given identity to verify the related identity. Use the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) to verify the identities \(\cosh ^{2} x=\frac{1+\cosh 2 x}{2}\) and \(\sinh ^{2} x=\frac{\cosh 2 x-1}{2}\).

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. According to the 2010 census, the U.S. population was 309 million with an estimated growth rate of \(0.8 \%\) / yr. a. Based on these figures, find the doubling time and project the population in 2050. b. Suppose the actual growth rates are just 0.2 percentage points lower and higher than \(0.8 \% / \mathrm{yr}(0.6 \% \text { and } 1.0 \%) .\) What are the resulting doubling times and projected 2050 populations? c. Comment on the sensitivity of these projections to the growth rate.

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{e^{2 x}}{4+e^{2 x}} d x$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\cos 2 x ;[0, \pi]$$

Find the area of the surface generated when the given curve is revolved about the \(y\) -axis. $$y=(3 x)^{1 / 3}, \text { for } 0 \leq x \leq \frac{8}{3}$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=4-x, y=2, \text { and } x=0$$

Sketch the following regions (if a figure is not given) and then find the area. The region bounded by \(y=x, y=1 / x, y=0,\) and \(x=2\)

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the anti- derivative method and the Fundamental Theorem of Calculus (Theorem 1). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=9-t^{2} \text { on }[0,4] ; s(0)=-2$$