Ch 2.6 - 89AE

Problem 4E Sketch the graph of a function that is continuous on an open interval (a ? , b?)but has neither an absolute maximum nor an absolute minimum value on (a? ,?? .)

Problem 89AE

Problem 88AE

Problem 90AE

Problem 91AE

Problem 92AE

a. Use the identity sin (a + h) = sin a cos h + cos a sin h with the fact that \(\lim \limits_{x \rightarrow 0} \sin x=0\) to prove that \(\lim \limits_{x \rightarrow a} \sin x=\sin a\), thereby establishing that sin x is continuous for all x. (Hint: Let h = x - a so that x = a + h and note that \(h \rightarrow 0\) as \(x \rightarrow a\).) b. Use the identity cos (a + h) = cos a cos h - sin a sin h with the fact that \(\lim _{x \rightarrow 0} \cos x=1\) to prove that \(\lim _{x \rightarrow a} \cos x=\cos a\).

Which of the following functions are continuous for all values in their domain? Justify your answers. a. a(t) = altitude of a skydiver t seconds after jumping from a plane b. n(t) = number of quarters needed to park in a metered parking space for t minutes c. T(t) = temperature t minutes after midnight in Chicago on January 1 d. p(t) = number of points scored by a basketball player after t minutes of a basketball game

Give the three conditions that must be satisfied by a function to be continuous at a point.

What does it mean for a function to be continuous on an interval?

Complete the following sentences. a. A function is continuous from the left at a if ________. b. A function is continuous from the right at a if ________.

Describe the points (if any) at which a rational function fails to be continuous.

What is the domain of \(f(x)=e^{x} / x\) and where is f continuous?

Explain in words and pictures what the Intermediate Value Theorem says.

Determine the points at which the following functions f have discontinuities. For each point state the conditions in the continuity checklist that are violated.

Determine the points at which the following functions f have discontinuities. For each point state the conditions in the continuity checklist that are violated.

Determine the points at which the following functions f have discontinuities. For each point state the conditions in the continuity checklist that are violated.

Determine the points at which the following functions f have discontinuities. For each point state the conditions in the continuity checklist that are violated.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(f(x)=\sqrt{x-2} ; \quad a=1\)

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(g(x)=\frac{1}{x-3} ; a=3\)

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text { if } x \neq 1 \\ 3 & \text { if } x=1 \end{array} ; a=1\right.\)

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(f(x)=\left\{\begin{array}{ll} \frac{x^{2}-4 x+3}{x-3} & \text { if } x \neq 3 \\ 2 & \text { if } x=3 \end{array} ; a=3\right.\)

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(f(x)=\frac{5 x-2}{x^{2}-9 x+20} ; a=4\)

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. \(f(x)=\left\{\begin{array}{ll} \frac{x^{2}-x}{x+1} & \text { if } x \neq-1 \\ 0 & \text { if } x=-1 \end{array}\right.\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(p(x)=4 x^{5}-3 x^{2}+1\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(g(x)=\frac{3 x^{2}-6 x+7}{x^{2}+x+1}\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(f(x)=\frac{x^{5}+6 x+17}{x^{2}-9}\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(s(x)=\frac{x^{2}-4 x+3}{x^{2}-1}\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(f(x)=\frac{1}{x^{2}-4}\)

Use Theorem 2.10 to determine the intervals on which the following functions are continuous. \(f(t)=\frac{t+2}{t^{2}-4}\)

Evaluate the following limits and justify your answers. \(\lim \limits_{x \rightarrow 0}\left(x^{8}-3 x^{6}-1\right)^{40}\)

Evaluate the following limits and justify your answers. \(\lim \limits_{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4}\)

Evaluate the following limits and justify your answers. \(\lim \limits_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4}\)

Evaluate the following limits and justify your answers. \(\lim \limits_{x \rightarrow \infty}\left(\frac{2 x+1}{x}\right)^{3}\)

Determine the intervals of continuity for the following functions. The graph of Exercise 9

Determine the intervals of continuity for the following functions. The graph of Exercise 10

Determine the intervals of continuity for the following functions. The graph of Exercise 11

Determine the intervals of continuity for the following functions. The graph of Exercise 12

Let \(f(x)=\left\{\begin{array}{ll} x^{2}+3 x & \text { if } x \geq 1 \\ 2 x & \text { if } x<1 \end{array}\right.\) a. Use the continuity checklist to show that f is not continuous at 1. b. Is f continuous from the left or right at 1? c. State the interval(s) of continuity.

Let \(f(x)=\left\{\begin{array}{ll} x^{3}+4 x+1 & \text { if } x \leq 0 \\ 2 x^{3} & \text { if } x>0 \end{array}\right.\) a. Use the continuity checklist to show that f is not continuous at 0. b. Is f continuous from the left or right at 0? c. State the interval(s) of continuity.

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(f(x)=\sqrt{2 x^{2}-16}\)

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(g(x)=\sqrt{x^{4}-1}\)

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(f(x)=\sqrt[3]{x^{2}-2 x-3}\)

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(f(t)=\left(t^{2}-1\right)^{3 / 2}\)

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(f(x)=(2 x-3)^{2 / 3}\)

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. \(f(z)=(z-1)^{3 / 4}\)

Determine the following limits and justify your answers. \(\lim \limits_{x \rightarrow 2} \sqrt{\frac{4 x+10}{2 x-2}}\)

Determine the following limits and justify your answers. \(\lim \limits_{x \rightarrow-1}\left(x^{2}-4+\sqrt[3]{x^{2}-9}\right)\)

Determine the following limits and justify your answers. \(\lim \limits_{x \rightarrow 3}\left(\sqrt{x^{2}+7}\right)\)

Determine the following limits and justify your answers. \(\lim \limits_{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}\)

Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. \(f(x)=\csc x ; \quad \lim \limits_{x \rightarrow \pi / 4} f(x) ; \quad \lim \limits_{x \rightarrow 2 \pi^{-}} f(x)\)

Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. \(f(x)=e^{\sqrt{x}} ; \quad \lim \limits_{x \rightarrow 4} f(x) ; \quad \lim \limits_{x \rightarrow 0^{+}} f(x)\)

Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. \(f(x)=\frac{1+\sin x}{\cos x} ; \quad \lim \limits_{x \rightarrow \pi / 2^{-}} f(x) ; \quad \lim \limits_{x \rightarrow 4 \pi / 3} f(x)\)

Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. \(f(x)=\frac{\ln x}{\sin ^{-1} x} ; \quad \lim \limits_{x \rightarrow 1^{-}} f(x)\)

Suppose $5000 is invested in a savings account for 10 years (120 months), with an annual interest rate of r, compounded monthly. The amount of money in the account after 10 years is \(A(r)=5000(1+r / 12)^{120}\). a. Use the Intermediate Value Theorem to show there is a value of r in (0, 0.08)—an interest rate between 0% and 8%—that allows you to reach your savings goal of $7000 in 10 years. b. Use a graph to illustrate your explanation in part (a); then, approximate the interest rate required to reach your goal.

You are shopping for a $150,000, 30-year (360-month) loan to buy a house. The monthly payment is \(m(r)=\frac{150,000(r / 12)}{1-(1+r / 12)^{-360}}\) where r is the annual interest rate. Suppose banks are currently offering interest rates between 6% and 8%. a. Use the Intermediate Value Theorem to show there is a value of r in (0.06, 0.08)—an interest rate between 6% and 8%—that allows you to make monthly payments of $1000 per month. b. Use a graph to illustrate your explanation to part (a). Then determine the interest rate you need for monthly payments of $1000.

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval (a, b). b. Use a graphing utility to find all the solutions to the equation on (a, b). c. Illustrate your answers with an appropriate graph. \(2 x^{3}+x-2=0 ; \quad(-1,1)\)

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval (a, b). b. Use a graphing utility to find all the solutions to the equation on (a, b). c. Illustrate your answers with an appropriate graph. \(\sqrt{x^{4}+25 x^{3}+10}=5 ; \quad(0,1)\)

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval (a, b). b. Use a graphing utility to find all the solutions to the equation on (a, b). c. Illustrate your answers with an appropriate graph. \(x^{3}-5 x^{2}+2 x=-1 ; \quad(-1,5)\)

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval (a, b). b. Use a graphing utility to find all the solutions to the equation on (a, b). c. Illustrate your answers with an appropriate graph. \(-x^{5}-4 x^{2}+2 \sqrt{x}+5=0 ; \quad(0,3)\)

Determine whether the following statements are true and give an explanation or counterexample. a. If a function is left-continuous and right-continuous at a, then it is continuous at a. b. If a function is continuous at a, then it is left-continuous and right-continuous at a. c. If a < b and \(f(a) \leq L \leq f(b)\), then there is some value of c between a and b for which f(c) = L. d. Suppose f is continuous on [a, b]. Then there is a point c in (a, b) such that f(c) = [f(a) + f(b)]/2.

Prove that the absolute value function |x| is continuous for all values of x. (Hint: Using the definition of the absolute value function, compute \(\lim \limits_{x \rightarrow 0^{-}}|x|\) and \(\lim \limits_{x \rightarrow 0^{+}}|x|\).)

Use the continuity of the absolute value function (Exercise 56) to determine the interval(s) on which the following functions are continuous. \(f(x)=\left|x^{2}+3 x-18\right|\)

Use the continuity of the absolute value function (Exercise 56) to determine the interval(s) on which the following functions are continuous. \(g(x)=\left|\frac{x+4}{x^{2}-4}\right|\)

Use the continuity of the absolute value function (Exercise 56) to determine the interval(s) on which the following functions are continuous. \(h(x)=\left|\frac{1}{\sqrt{x}-4}\right|\)

Use the continuity of the absolute value function (Exercise 56) to determine the interval(s) on which the following functions are continuous. \(h(x)=\left|x^{2}+2 x+5\right|+\sqrt{x}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow \pi} \frac{\cos ^{2} x+3 \cos x+2}{\cos x+1}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 5 \pi / 2} \frac{\sin ^{2} x+6 \sin x+5}{\sin ^{2} x-1}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow \pi / 2} \frac{\sin x-1}{\sqrt{\sin x}-1}\)

Evaluate the following limits. \(\lim \limits_{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0^{+}} \cot x\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow \infty} \frac{\tan ^{-1} x}{x}\)

Evaluate the following limits. \(\lim \limits_{t \rightarrow \infty} \frac{\cos t}{e^{3 t}}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1^{-}} \frac{x}{\ln x}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0^{+}} \frac{x}{\ln x}\)

The graph of the sawtooth function \(y=x-\lfloor x\rfloor\), where \(\lfloor x\rfloor\) is the greatest integer function or floor function (Exercise 31, Section 2.2), was obtained using a graphing utility (see figure). Identify any inaccuracies appearing in the graph and then plot an accurate graph by hand.

Graph the function \(f(x)=\frac{\sin x}{x}\) using a graphing window of \([-\pi, \pi] \times[0,2]\). a. Sketch a copy of the graph obtained with your graphing device and describe any inaccuracies appearing in the graph. b. Sketch an accurate graph of the function. Is f continuous at 0?

a. Sketch the graph of a function that is not continuous at 1, but is defined at 1. b. Sketch the graph of a function that is not continuous at 1, but has a limit at 1.

Determine the value of the constant a for which the function \(f(x)=\left\{\begin{array}{ll} \frac{x^{2}+3 x+2}{x+1} & \text { if } x \neq-1 \\ a & \text { if } x=-1 \end{array}\right.\) is continuous at -1.

Let \(g(x)=\left\{\begin{array}{ll} x^{2}+x & \text { if } x<1 \\ a & \text { if } x=1 \\ 3 x+5 & \text { if } x>1 \end{array}\right.\) a. Determine the value of a for which g is continuous from the left at 1. b. Determine the value of a for which g is continuous from the right at 1. c. Is there a value of a for which g is continuous at 1? Explain.

Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. \(x^{3}+10 x^{2}-100 x+50=0 ;(-20,10)\)

Use the Intermediate Value Theorem to verify that the following equations have three solutions on the given interval. Use a graphing utility to find the approximate roots. \(70 x^{3}-87 x^{2}+32 x-3=0 ;(0,1)\)

Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see the figure). Consider \(0 \leq t \leq 60\).

Assume you invest $250 at the end of each year for 10 years at an annual interest rate of r. The amount of money in your account after 10 years is \(A=\frac{250\left[(1+r)^{10}-1\right]}{r}\). Assume your goal is to have $3500 in your account after 10 years. a. Use the Intermediate Value Theorem to show that there is an interest rate r in (0.01, 0.10)—between 1% and 10%—that allows you to reach your financial goal. b. Use a calculator to estimate the interest rate required to reach your financial goal.

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 A.M. on Friday morning and let g(t) be your distance from the car t hours after 7 A.M. on Sunday morning. a. Evaluate f(0), f(2), g(0), and g(2). b. Let h(t) = f(t) – g(t). Find h(0) and h(2). c. Use the Intermediate Value Theorem to show that there is some point along the trail that you will pass at exactly the same time of morning on both days.

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day, the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation.)

Does continuity of |f| imply continuity of f? Let \(g(x)=\left\{\begin{array}{ll} 1 & \text { if } x \geq 0 \\ -1 & \text { if } x<0 \end{array}\right.\) a. Write a formula for |g(x)|. b. Is g continuous at x = 0? Explain. c. Is |g| continuous at x = 0? Explain. d. For any function f, if |f| is continuous at a, does it necessarily follow that f is continuous at a? Explain.

The discontinuities in graphs (a) and (b) are removable discontinuities because they disappear if we redefine f at a so that \(f(a)=\lim \limits_{x \rightarrow a} f(x)\). The function in graph (c) has a jump discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph (d) is an infinite discontinuity because the function has a vertical asymptote at a. Is the discontinuity at a in graph (c) removable? Explain.

The discontinuities in graphs (a) and (b) are removable discontinuities because they disappear if we redefine f at a so that \(f(a)=\lim \limits_{x \rightarrow a} f(x)\). The function in graph (c) has a jump discontinuity because left and right limits exist at a but are unequal. The discontinuity in graph (d) is an infinite discontinuity because the function has a vertical asymptote at a. Is the discontinuity at a in graph (d) removable?

Show that the following functions have a removable discontinuity at the given point. \(f(x)=\frac{x^{2}-7 x+10}{x-2} ; x=2\)

Show that the following functions have a removable discontinuity at the given point. \(g(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{1-x} & \text { if } x \neq 1 \\ 3 & \text { if } x=1 \end{array} ; x=1\right.\)

a. Does the function f(x) = x sin (1/x) have a removable discontinuity at x = 0? b. Does the function g(x) = sin (1/x) have a removable discontinuity at x = 0?