Chapter 6

Give two examples of processes that are modeled by exponential growth.

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) \(y=2 x+1\) on [1,5] (Use calculus.)

Evaluate the following derivatives. \(\left.\frac{d}{d x}\left(x \ln x^{3}\right)\right|_{x=1}\)

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=8 \sqrt{x} \text { on }[9,20]$$

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. $$y=\left(1+x^{2}\right)^{-1}, y=0, x=0, \text { and } x=2$$

Give two examples of processes that are modeled by exponential decay.

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) \(y=4-3 x\) on [-3,2] (Use calculus.)

On what interval is the formula \(d / d x\left(\tanh ^{-1} x\right)=1 /\left(x^{2}-1\right)\) valid?

Evaluate the following derivatives. \(\frac{d}{d x}(\ln (\ln x))\)

Explain why you integrate in the vertical direction (parallel to the acceleration due to gravity) rather than the horizontal direction to find the force on the face of a dam.

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=x^{3} \text { on }[0,1]$$

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. $$y=6-x, y=0, x=2, \text { and } x=4$$

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

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=6-2 t \text { on } 0 \leq t \leq 6$$

Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=100+10.5 t, g(t)=100 e^{t / 10}$$

When evaluating the definite integral \(\int_{6}^{8} \frac{d x}{16-x^{2}},\) why must you choose the antiderivative \(\frac{1}{4} \operatorname{coth}^{-1} \frac{x}{4}\) rather than \(\frac{1}{4} \tanh ^{-1} \frac{x}{4} ?\)

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) \(y=-8 x-3\) on [-2,6] (Use calculus.)

Evaluate the following derivatives. \(\frac{d}{d x}(\sin (\ln x))\)

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=1+\sin x ; \text { for } 0 \leq x \leq \pi$$

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

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. \(y=3 x, y=3,\) and \(x=0\) (Use integration and check your answer using the volume formula for a cone.)

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the curve \(y=\sqrt{\cos x}\) and the \(x\) -axis on \([-\pi / 2, \pi / 2],\) and whose cross sections through the solid perpendicular to the \(x\) -axis are isosceles right triangles with a horizontal leg in the \(x y\) -plane and a vertical leg above the \(x\) -axis.

Sketch the region and find its area. The region bounded by \(y=2(x+1), y=3(x+1),\) and \(x=4\)

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=10 \sin 2 t \text { on } 0 \leq t \leq 2 \pi$$

Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=2200+400 t, g(t)=400 \cdot 2^{1 / 20}$$

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) $$y=\frac{1}{2}\left(e^{x}+e^{-x}\right) \text { on }[-\ln 2, \ln 2]$$

How does the graph of the catenary \(y=a \cosh (x / a)\) change as \(a>0\) increases?

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

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=1+x^{3} ; \text { for } 0 \leq x \leq 1$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\sqrt{4 x+6} \text { on }[0,5]$$

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. \(y=1-x^{2}, x=0,\) and \(y=0,\) in the first quadrant

Sketch the region and find its area. The region bounded by \(y=\cos x\) and \(y=\sin x\) between \(x=\pi / 4\) and \(x=5 \pi / 4\)

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 population of a town with a 2010 population of 90,000 grows at a rate of \(2.4 \% /\) yr. In what year will the population double its initial value (to \(180,000) ?\)

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=t^{2}-6 t+8 \text { on } 0 \leq t \leq 5$$

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) $$y=\frac{1}{3} x^{3 / 2} \text { on }[0,60]$$

Verify each identity using the definitions of the hyperbolic functions. $$\tanh x=\frac{e^{2 x}-1}{e^{2 x}+1}$$

Evaluate the following derivatives. \(\frac{d}{d x}\left((\ln 2 x)^{-5}\right)\)

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=2-x / 2 ; \text { for } 0 \leq x \leq 2$$

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

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. $$y=x^{3}-x^{8}+1, y=1$$

Use the general slicing method to find the volume of the following solids. The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares.

Sketch the region and find its area. The region bounded by \(y=e^{x}, y=e^{-2 x},\) and \(x=\ln 4\)

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 population of Clark County, Nevada, was 2 million in 2013. Assuming an annual growth rate of 4.5\%/yr, what will the county population be in \(2020 ?\)

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=-t^{2}+5 t-4 \text { on } 0 \leq t \leq 5$$

Find the arc length of the following curves on the given interval by integrating with respect to \(x\) $$y=3 \ln x-\frac{x^{2}}{24} \text { on }[1,6]$$

Verify each identity using the definitions of the hyperbolic functions. $$\tanh (-x)=-\tanh x$$

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

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=5 e^{-2 x} ; \text { for } 0 \leq x \leq 4$$

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. $$y=\sqrt{x}, y=0, \text { and } x=1$$