Chapter 6

In terms of relative growth rate, what is the defining property of exponential growth?

State the definition of the hyperbolic cosine and hyperbolic sine functions.

Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line.

What is the area of the curved surface of a right circular cone of radius 3 and height 4?

Explain the steps required to find the length of a curve \(y=f(x)\) between \(x=a\) and \(x=b\)

What are the domain and range of \(\ln x ?\)

Suppose a \(1-\mathrm{m}\) cylindrical bar has a constant density of \(1 \mathrm{g} / \mathrm{cm}\) for its left half and a constant density \(2 \mathrm{g} / \mathrm{cm}\) for its right half. What is its mass?

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly twice on \((-\infty, \infty) .\) Explain how to use integration to find the area of the region bounded by the two curves.

Suppose a cut is made through a solid object perpendicular to the \(x\) -axis at a particular point \(x .\) Explain the meaning of \(A(x)\).

Give two pieces of information that may be used to formulate an exponential growth or decay function.

Suppose the velocity of an object moving along a line is positive. Are displacement and distance traveled equal? Explain.

Sketch the graphs of \(y=\cosh x, y=\sinh x,\) and \(y=\tanh x\) (include asymptotes), and state whether each function is even, odd, or neither.

A frustum of a cone is generated by revolving the graph of \(y=4 x\) on the interval [2,6] about the \(x\) -axis. What is the area of the surface of the frustum?

Explain the steps required to find the length of a curve \(x=g(y)\) between \(y=c\) and \(y=d\)

Fill in the blanks: A region \(R\) is revolved about the \(y\) -axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to __________ or using the shell method and integrating with respect to __________.

A solid has a circular base and cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly three times on \((-\infty, \infty) .\) How is integration used to find the area of the region bounded by the two curves?

Explain the meaning of doubling time.

Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to \(x\) that gives the length of the following curves on the given interval. $$y=x^{3}+2 \text { on }[-2,5]$$

Given the velocity function \(v\) of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

What is the fundamental identity for hyperbolic functions?

Suppose \(f\) is positive and differentiable on \([a, b] .\) The curve \(y=f(x)\) on \([a, b]\) is revolved about the \(x\) -axis. Explain how to find the area of the surface that is generated.

Fill in the blanks: A region \(R\) is revolved about the \(x\) -axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to __________ or using the shell method and integrating with respect to __________.

Evaluate \(\int 4^{x} d x\).

How much work is required to move an object from \(x=0\) to \(x=5\) (measured in meters) in the presence of a constant force of 5 N acting along the \(x\) -axis?

The region bounded by the curves \(y=2 x\) and \(y=x^{2}\) is revolved about the \(x\) -axis. Give an integral for the volume of the solid that is generated.

Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to \(x\).

Explain the meaning of half-life.

Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to \(x\) that gives the length of the following curves on the given interval. $$y=2 \cos 3 x \text { on }[-\pi, \pi]$$

How are the derivative formulas for the hyperbolic functions and the trigonometric functions alike? How are they different?

Explain how to use definite integrals to find the net change in a quantity, given the rate of change of that quantity.

Suppose \(g\) is positive and differentiable on \([c, d] .\) The curve \(x=g(y)\) on \([c, d]\) is revolved about the \(y\) -axis. Explain how to find the area of the surface that is generated.

What is the inverse function of \(\ln x,\) and what are its domain and range?

Why is integration used to find the work done by a variable force?

The region bounded by the curves \(y=2 x\) and \(y=x^{2}\) is revolved about the \(y\) -axis. Give an integral for the volume of the solid that is generated.

Given the rate of change of a quantity \(Q\) and its initial value \(Q(0)\) explain how to find the value of \(Q\) at a future time \(t \geq 0\)

Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to \(x\) that gives the length of the following curves on the given interval. $$y=e^{-2 x} \text { on }[0,2]$$

Express \(\sinh ^{-1} x\) in terms of logarithms.

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=3 x+4 \text { on }[0,6]$$

Why is integration used to find the work required to pump water out of a tank?

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. $$y=x-x^{2}, y=0$$

Express \(3^{x}, x^{\pi},\) and \(x^{\sin x}\) using the base \(e\).

Why is the disk method a special case of the general slicing method?

How are the rate constant and the half-life related?

What is the result of integrating a population growth rate between times \(t=a\) and \(t=b,\) where \(b>a ?\)

What is the domain of \(\operatorname{sech}^{-1} x ?\) How is \(\operatorname{sech}^{-1} x\) defined in terms of the inverse hyperbolic cosine?

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=12-3 x \text { on }[1,3]$$

Why is integration used to find the total force on the face of a dam?

Evaluate \(\frac{d}{d x}\left(3^{x}\right)\).

The region \(R\) bounded by the graph of \(y=f(x) \geq 0\) and the \(x\) -axis on \([a, b]\) is revolved about the line \(y=-2\) to form a solid of revolution whose cross sections are washers. What are the inner and outer radii of the washer at a point \(x\) in \([a, b] ?\)