Chapter 11

Fill in each blank with the appropriate word or phrase. The slope of a line drawn tangent to the graph of any function \(f(x)\) can be found by taking the limit of the _____ quotient for \(f,\) as \(h\) approaches zero.

Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary. $$\lim _{x \rightarrow c} g(x)=L$$ means that values of \(g(x)\) can be made ____________ to \(L,\) by taking values of ____________ sufficiently close to ____________

Fill in each blank with the appropriate word or phrase. In order to obtain a better approximation for the area under a curve, we can use narrower (and thus more) _____ in our computation.

Fill in each blank with the appropriate word or phrase. As we let the number of approximating rectangles increase without bound, the resulting limit gives \(a(n)\) _____ value for the area under the curve.

Discuss/Explain the relationship between the difference quotient and the formula for finding the slope of a line given two points.

Discuss/Explain the relationship between the horizontal asymptote of a rational function \(f\), and \(\lim _{x \rightarrow \infty} f(x)\).

Discuss/Explain why \(\lim _{x \rightarrow 3} g(x)=5\) for \(g(x)=\) \(\frac{x^{2}-x-6}{x-3},\) but \(g(3) \neq 5 .\) Can we redefine \(g(x)\) to create a piecewise-defined function where \(\lim _{x \rightarrow 3} g(x)=g(3) ?\)

Discuss/Explain how the three different types of discontinuities appear on the graph of a function.

Discuss/Explain why \(\lim _{x \rightarrow 1} f(x)\) does not exist for \(f(x)=\frac{x^{2}-x}{\sqrt{(x-1)^{2}}},\) even though the left-hand limit and the right-hand limit do exist.

Two model rockets are launched at a gathering of the National Association of Rocketry (NAR: www.nar.org). Frank's Apollo II motor burns out at a height of \(500 \mathrm{m},\) at which point the rocket has a velocity of 88.2 meters per second (m/sec). His rocket's height in meters, \(t\) sec after engine burnout, is given by \(f(t)=500+88.2 t-4.9 t^{2}\) Gwen's Icarus Alpha motor burns out at a height of \(600 \mathrm{m}\) at which point the rocket has a velocity of \(78.4 \mathrm{m} / \mathrm{sec} .\) Her rocket's height in meters, \(t\) sec after burnout, is given by \(g(t)=600+78.4 t-4.9 t^{2}\). Find the limit of the difference quotient for \(f,\) to obtain a function \(f(t)\) that represents the instantaneous velocity at time \(t\).

Write each of the following statements using limit notation. As \(n\) approaches \(\infty, V_{n}\) approaches \(\frac{4}{3} \pi r^{3}\)

Write each of the following statements using limit notation. As \(n\) approaches \(\infty, A_{n}\) approaches \(\frac{a}{3} x^{3}+\frac{b}{2} x^{2}+c x+d\)

Two model rockets are launched at a gathering of the National Association of Rocketry (NAR: www.nar.org). Frank's Apollo II motor burns out at a height of \(500 \mathrm{m},\) at which point the rocket has a velocity of 88.2 meters per second (m/sec). His rocket's height in meters, \(t\) sec after engine burnout, is given by \(f(t)=500+88.2 t-4.9 t^{2}\) Gwen's Icarus Alpha motor burns out at a height of \(600 \mathrm{m}\) at which point the rocket has a velocity of \(78.4 \mathrm{m} / \mathrm{sec} .\) Her rocket's height in meters, \(t\) sec after burnout, is given by \(g(t)=600+78.4 t-4.9 t^{2}\). Find the limit of the difference quotient for \(g\), to obtain a function \(g(t)\) that represents the instantaneous velocity at time \(t\).

Write each of the following statements using limit notation. As \(t\) approaches \(-\infty, e^{/ n}\) approaches 0

Write each of the following statements using limit notation. As \(t\) approaches \(-\infty, \sin [g(t)]\) approaches 1

Two model rockets are launched at a gathering of the National Association of Rocketry (NAR: www.nar.org). Frank's Apollo II motor burns out at a height of \(500 \mathrm{m},\) at which point the rocket has a velocity of 88.2 meters per second (m/sec). His rocket's height in meters, \(t\) sec after engine burnout, is given by \(f(t)=500+88.2 t-4.9 t^{2}\) Gwen's Icarus Alpha motor burns out at a height of \(600 \mathrm{m}\) at which point the rocket has a velocity of \(78.4 \mathrm{m} / \mathrm{sec} .\) Her rocket's height in meters, \(t\) sec after burnout, is given by \(g(t)=600+78.4 t-4.9 t^{2}\). Use the result from Exercise 7 to find the maximum height of Frank's rocket. This occurs when \(v=0\).

Write each of the following statements using limit notation. As \(x\) increases without bound, \(\cos \left(\frac{1}{x}\right)\) approaches 1

Write each of the following statements using limit notation. As \(x\) decreases without bound, \(\frac{5 x^{2}-3}{2 x^{2}-x-1}\) approaches \(\frac{5}{2}\)

To replicate Galileo's famous test as to whether the velocity of a falling body depends on its weight, a science class is dropping bowling balls of different weights from an 11 -story building onto the lawn below. The ball's height in meters, \(t\) sec after it is, released, is modeled by \(d(t)=-4.9 t^{2}+44.1 .\) Find the limit of the difference quotient for \(d\), to obtain a function \(d(t)\) that represents the instantaneous velocity of the bowling ball at time \(t.\)

If an \(n\) -sided regular polygon is inscribed in a circle of radius \(r\), its perimeter is given by \(P=2 n r \sin \left(\frac{\pi}{n}\right) .\) For a circle with radius \(r=50 \mathrm{mm},\) determine the number of sides needed (to the nearest 25 ) for the perimeter of the polygon to approximate the circumference of the circle correct to two decimal places when rounded.

A rock climber's carabineer falls off her harness \(256 \mathrm{ft}\) above the floor of the Grand Canyon. It's height in feet, \(t\) sec after it falls, can be modeled by \(d(t)=-16 t^{2}+256 .\) Find the limit of the difference quotient for \(d\), to obtain a function \(d l(t)\) that represents the instantaneous velocity of the "biner" at time \(t.\)

One of the most famous and useful numbers in all of mathematics is the one defined as \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e\) Determine the smallest \(x\) to (the nearest 250 ) that can be used to approximate the value of \(e\) to three decimal places.

Find the limit of the difference quotient for each function \(f(x)\) given, to obtain a function \(f(x)\) that represents the instantaneous rate of change at \(x\) for each function. $$f(x)=\frac{1}{2} x+5$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-3} 2 x^{2}-5 x+3$$

Write each of the following statements using limit notation. As \(t\) approaches \(5, s_{t}\) approaches \(5 r\)

Find the limit of the difference quotient for each function \(f(x)\) given, to obtain a function \(f(x)\) that represents the instantaneous rate of change at \(x\) for each function. $$f(x)=x^{2}-3 x$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-2} 3 x^{2}-5 x-2$$

Write each of the following statements using limit notation. As \(t\) approaches \(-3, d_{t}\) approaches \(\sqrt{9+r^{2}}\)

Find the limit of the difference quotient for each function \(f(x)\) given, to obtain a function \(f(x)\) that represents the instantaneous rate of change at \(x\) for each function. $$f(x)=x^{3}$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow 5} \sqrt{3 x-19}$$

Write each of the following statements using limit notation. As \(x\) approaches \(a, \tan ^{-1}[g(x)]\) approaches \(\frac{\pi}{3}\)

Find the limit of the difference quotient for each function \(f(x)\) given, to obtain a function \(f(x)\) that represents the instantaneous rate of change at \(x\) for each function. $$f(x)=\frac{1}{x}$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-6} \sqrt{x^{2}+7 x}$$

Write each of the following statements using limit notation. As \(x\) approaches \(b, \csc ^{2}[f(x)]\) approaches \(\frac{4}{3}\)

The population of a small town can be modeled by the function \(p(t)=1.2 \sqrt{t}+40,\) where \(p\) is measured in thousands and \(t\) is the number of years after \(2008 .\) Find the limit of the difference quotient for \(p\) to obtain a function \(p(t)\) that represents the instantaneous rate of change of population at time \(t.\)

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow 2} \frac{x^{2}}{5 x-2}$$

Write each of the following statements using limit notation. As \(x \rightarrow-3, \frac{x+3}{x^{2}-9}\) approaches \(-\frac{1}{6}\)

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow 8} \frac{2 x-5}{x^{2}-5 x}$$

Write each of the following statements using limit notation. As \(x \rightarrow \pi, \frac{\sin x}{\cos \left(\frac{x}{2}\right)}\) approaches 2.

The number of bacteria in a person's body, after they begin a regimen of antibiotics, can be modeled by the function \(b(t)=6-\sqrt{t},\) where \(b\) is measured in tens of thousands and \(t\) is the number of hours after the first dose. Find the limit of the difference quotient for \(b\) to obtain a function \(b(t)\) that represents the instantaneous rate of change of number of bacteria at time \(t.\)

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-1} \frac{x+1}{x^{2}-1}$$

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$p(x)=\cos x+\sin \left(\frac{3 x}{2}\right) ; x \rightarrow \pi$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-2} \frac{4-x^{2}}{x^{2}+5 x+6}$$

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$q(x)=\tan x+3 \sin 2 x ; x \rightarrow \frac{\pi}{4}$$

Find the limit of the difference quotient of the given function to obtain a function that represents the slope of a line drawn tangent to the curve at \(x.\) $$f(x)=\frac{2}{x-1}$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow-5} \sqrt{x^{2}-6 x}$$

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$v(x)=\frac{\cos \left(\frac{\pi}{x}\right)}{\sin (\pi x)} ; x \rightarrow 2$$

Find the limit of the difference quotient of the given function to obtain a function that represents the slope of a line drawn tangent to the curve at \(x.\) $$g(x)=\frac{3}{x+2}$$

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow 6} \sqrt{3 x-5}$$

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$w(x)=\frac{\tan (x-2)}{x-2} ; x \rightarrow 2$$