?Absolute Value and Continuity(a) Show that the absolute value function \(F(x)=

Write an equation that expresses the fact that a function \(f\) is continuous at the number 4. ________________ Equation Transcription: Text Transcription: f

If \(f\) is continuous on \((-\infty, \infty)\), what can you say about its graph? ________________ Equation Transcription: Text Transcription: F (?infinity, infinity)

(a) From the given graph of \(f\), state the numbers at which \(f\) is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether \(f\) is continuous from the right, or from the left,or neither. Equation Transcription: Text Transcription: f

From the given graph of \(g\), state the numbers at which \(g\) is discontinuous and explain why. ________________ Equation Transcription: Text Transcription: g

The graph of a function \(f\) is given. (a) At what numbers \(a\) does \(\lim _{x \rightarrow a} f(x)\) not exist? (b) At what numbers \(a\) is \(f\) not continuous? (c) At what numbers \(a\) does \(\lim _{x \rightarrow a} f(x)\) exist but \(f\) is not continuous at \(a\)? Equation Transcription: Text Transcription: f lim_x right arrow a f(x) a

The graph of a function \(f\) is given. (a) At what numbers \(a\) does \(\lim _{x \rightarrow a} f(x)\) not exist? (b) At what numbers \(a\) is \(f\) not continuous? (c) At what numbers \(a\) does \(\lim _{x \rightarrow a} f(x)\) exist but \(f\) is not continuous at \(a\)? Equation Transcription: Text Transcription: f lim_x right arrow a f(x) a

Sketch the graph of a function \(f\) that is defined on \(\mathbb{R}\) and continuous except for the stated discontinuities. Removable discontinuity at \(-2\), infinite discontinuity at \(2\) Equation Transcription: Text Transcription: f R -2 2

Sketch the graph of a function \(f\) that is defined on \(\mathbb{R}\) and continuous except for the stated discontinuities. Jump discontinuity at \(-3\), removable discontinuity at \(4\) Equation Transcription: Text Transcription: f R -3 4

Sketch the graph of a function \(f\) that is defined on \(\mathbb{R}\) and continuous except for the stated discontinuities. Discontinuities at \(0\) and \(3\), but continuous from the right at \(0\) and from the left at \(3\) Equation Transcription: Text Transcription: f R 3 0

Sketch the graph of a function \(f\) that is defined on \(\mathbb{R}\) and continuous except for the stated discontinuities. Continuous only from the left at \(-1\), not continuous from the left or right at \(3\) ________________ Equation Transcription: Text Transcription: f R 3 -1

The toll \(T\) charged for driving on a certain stretch of a toll road is \(\$ 5\) except during rush hours (between \(7 \mathrm{AM}\) and \(10 \mathrm{AM}\) and between 4 PM and 7 PM) when the toll is \(\$ 7\). (a) Sketch a graph of \(T\) as a function of the time \(t\), measured in hours past midnight. (b) Discuss the discontinuities of this function and their significance to someone who uses the road. Equation Transcription: Text Transcription: T $5 $7 t

Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(f(x)=3 x^{2}+(x+2)^{5}, \quad a=-1\) ________________ Equation Transcription: ???? Text Transcription: a f(x)=3x^2+(x+2)^5, a= -1

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(g(t)=\frac{t^{2}+5 t}{2 t+1}, \quad a=2\) ________________ Equation Transcription: ???? Text Transcription: a g(t)=t^2 + 5t/2t + 1, a=2

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. \(p(v)=2 \sqrt{3 v^2+1}, \quad a=1\)

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(f(r)=\sqrt[3]{4 r^{2}-2 r+7}, \quad a=-2\) ________________ Equation Transcription: ???? Text Transcription: a f(r)=cube root 4r^2-2r+7, a= -2

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(f(x)=x+\sqrt{x-4}, \quad[4, \infty)\) ________________ Equation Transcription: Text Transcription: f(x)=x+square root x-4, [4, infinity)

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(g(x)=\frac{x-1}{3 x+6}, \quad(-\infty,-2)\) ________________ Equation Transcription: Text Transcription: g(x)=x-1/3x+6, (-infinity,-2)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x)=\frac{1}{x+2}\) \(a=-2\) ________________ Equation Transcription: Text Transcription: a f(x)=1/x+2 a=-2

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x)=\left\{\begin{array}{ll}\frac{1}{x+2} & \text { if } x \neq-2 \\1 & \text { if } x=-2\end{array} \quad a=-2\right.\)

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x)=\left\{\begin{array}{ll}x+3 & \text { if } x \leqslant-1 \\2^{x} & \text { if } x>-1\end{array} \quad a=-1\right.\)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x)=\left\{\begin{array}{ll}\frac{x^{2}-x}{x^{2}-1} & \text { if } x \neq 1 \\ 1 & \text { if } x=1\end{array} \quad a=1\right.\) ________________ Equation Transcription Text Transcription: a f(x)={_1 if x not = 1^x^2-x/x^2-1 if x not = 1 a=1

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x)=\left\{\begin{array}{ll}\cos x & \text { if } x<0 \\0 & \text { if } x=0 \\1-x^{2} & \text { if } x>0\end{array} \quad a=0\right.\)

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x)=\left\{\begin{array}{ll}\frac{2 x^{2}-5 x-3}{x-3} & \text { if } x \neq 3 \\ 6 & \text { if } x=3\end{array} \quad a=3\right.\) ________________ Equation Transcription: Text Transcription: a f(x)={_6 if x=3^2x^2-5x-3/x-3 if x not = 3 a=3

(a) Show that f has a removable discontinuity at x=3. (b) Redefine f(3) so that f is continuous at x=3 (and thus the discontinuity is "removed"). \(f(x)=\frac{x-3}{x^2-9}\)

(a) Show that \(f\) has a removable discontinuity at \(x=3\) (b) Redefine \(f(3)\) so that \(f\) is continuous at \(x=3\) (and thus the discontinuity is "removed"). \(f(x)=\frac{x^{2}-7 x+12}{x-3}\) Equation Transcription: Text Transcription: f x=3 f(3) f x=3 f(x)=x^2-7x + 12 over x-3

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(f(x)=\frac{x^{2}}{\sqrt{x^{4}+2}}\) Equation Transcription: Text Transcription: f(x)=\frac x^2 \sqrt x^4+2

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(g(v)=\frac{3 v-1}{v^{2}+2 v-15}\) Equation Transcription: Text Transcription: g(v)=\frac3 v-1 v^2+2 v-15

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(h(t)=\frac{\cos \left(t^{2}\right)}{1-e^{t}}\) Equation Transcription: Text Transcription: h(t)=\frac\cos (t^2)1-e^t

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(B(u)=\sqrt{3 u-2}+\sqrt[3]{2 u-3}\)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(L(v)=v \ln \left(1-v^{2}\right)\) Equation Transcription: Text Transcription: L(v)=vln(1-v2)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(f(t)=e^{-t^{2}} \ln \left(1+t^{2}\right)\) Equation Transcription: Text Transcription: f(t)=e-t2ln (1+t2)

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(M(x)=\sqrt{1+\frac{1}{x}}\) Equation Transcription: Text Transcription: M(x)=\sqrt 1+\frac 1 x

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain \(g(t)=\cos ^{-1}\left(e^{t}-1\right)\)

Use continuity to evaluate the limit. \(\lim _{x \rightarrow 2} x \sqrt{20-x^{2}}\) Equation Transcription: Text Transcription: \lim _x \rightarrow 2 x \sqrt{20-x^2

Use continuity to evaluate the limit. \(\lim _{\theta \rightarrow \pi / 2} \sin (\tan (\cos \theta))\) Equation Transcription: Text Transcription: \lim _{\theta \rightarrow \pi / 2} \sin (\tan (\cos \theta))

Use continuity to evaluate the limit. \(\lim _{x \rightarrow 1} \ln \left(\frac{5-x^{2}}{1+x}\right)\) Equation Transcription: Text Transcription: \lim _x \rightarrow 1 \ln (\frac 5-x^2 1+x)

Use continuity to evaluate the limit. \(\lim _{x \rightarrow 4} 3^{\sqrt{x^{2}-2 x-4}}\) Equation Transcription: Text Transcription: \lim _x \rightarrow 4 3^\sqrt x^2-2 x-4

Locate the discontinuities of the function and illustrate by graphing. \(f(x)=\frac{1}{\sqrt{1-\sin x}}\)

Locate the discontinuities of the function and illustrate by graphing. \(y=\arctan \frac{1}{x}\) Equation Transcription: Text Transcription: y=arctan 1 over x

Show that \(f\) is continuous on \((-\infty, \infty)\). \(f(x)= \begin{cases}1-x^{2} & \text { if } x \leqslant 1 \\ \ln x & \text { if } x>1\end{cases}\) Equation Transcription: { Text Transcription: f (-\infty, \infty) f(x)= {1-x^{2} if x \leqslant 1 \ln x if x>1

Show that \(f\) is continuous on \((-\infty, \infty)\). \(f(x)= \begin{cases}\sin x & \text { if } x<\pi / 4 \\ \cos x & \text { if } x \geqslant \pi / 4\end{cases}\) Equation Transcription: { Text Transcription: f (-\infty, \infty) f(x)= \sin x if x<\pi / 4 \cos x if x \geqslant \pi / 4

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. \(f(x)= \begin{cases}x^{2} & \text { if } x<-1 \\ x & \text { if }-1 \leqslant x<1 \\ 1 / x & \text { if } x \geqslant 1\end{cases}\)

Find the numbers at which \(f\) is discontinuous. At which of these numbers is \(f\) continuous from the right, from the left, or neither? Sketch the graph of \(f\). \(f(x)= \begin{cases}2^{x} & \text { if } x \leqslant 1 \\ 3-x & \text { if } 1<x \leqslant 4 \\ \sqrt{x} & \text { if } x>4\end{cases}\) Equation Transcription: { Text Transcription: f f f f(x)= {2^x if x \leqslant 1 \\ 3-x if 1<x \leqslant 4\sqrt x if x>4

Find the numbers at which \(f\) is discontinuous. At which of these numbers is \(f\) continuous from the right, from the left, or neither? Sketch the graph of \(f\). \(f(x)= \begin{cases}x+2 & \text { if } x<0 \\ e^{x} & \text { if } 0 \leqslant x \leqslant 1 \\ 2-x & \text { if } x>1\end{cases}\) Equation Transcription: { Text Transcription: f f f f(x)= x+2 if x<0 \\ e^x if 0 \leqslant x \leqslant 1 \\ 2-x if x>1

The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is \(F(r)= \begin{cases}\frac{G M r}{R^{3}} & \text { if } r<R \\ \frac{G M}{r^{2}} & \text { if } r \geqslant R\end{cases}\) where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

For what value of the constant \(c\) is the function \(f\) continuous on \((-\infty, \infty)\)? \(f(x)= \begin{cases}c x^{2}+2 x & \text { if } x<2 \\ x^{3}-c x & \text { if } x \geqslant 2\end{cases}\) Find the values of and that make Equation Transcription: { Text Transcription: c f (-\infty, \infty) f(x)= \begin{cases}c x^{2}+2 x & \text { if } x<2 \\ x^{3}-c x & \text { if } x \geqslant 2

Find the values of and that make continuous everywhere. \(f(x)= \begin{cases}\frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 & \text { if } 2 \leqslant x<3 \\ 2 x-a+b & \text { if } x \geqslant 3\end{cases}\) Equation Transcription: { Text Transcription: f(x)= \frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 if 2 \leqslant x<3 2 x-a+b if x \geqslant 3

Suppose \(f \text { and } g\) are continuous functions such that \(g(2)=6\) and \(\lim _{x \rightarrow 2}[3 f(x)+f(x) g(x)]=36\). Find \(f(2)\). Equation Transcription: Text Transcription: f \text { and } g g(2)=6 \lim _{x \rightarrow 2}[3 f(x)+f(x) g(x)]=36 f(2)

Let \(f(x)=1 / x \text { and } g(x)=1 / x^{2}\). (a) Find \((f \circ g)(x)\) (b) Is \(f \circ g\) continuous everywhere? Explain. Equation Transcription: Text Transcription: f(x)=1 / x \text { and } g(x)=1 / x^{2} (f \circ g)(x) f \circ g

Which of the following functions \(f\) has a removable discontinuity at \(a\)? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \neq a\) and is continuous at \(a\). (a) \(f(x)=\frac{x^{4}-1}{x-1}, \quad a=1\) (b) \(f(x)=\frac{x^{3}-x^{2}-2 x}{x-2}, \quad a=2\) (c) \(f(x)=[\sin x], \quad a=\pi\) Equation Transcription: Text Transcription: f a g f x neq a a f(x)=x^4-1/x-1, a=1 f(x)=x^3-x^2-2x/x-2, a=2 f(x)=[sin x] a=pi

Suppose that a function \(f\) is continuous on \({[0,1]}\) except at \(0.25\) and that \(f(0)=1 \text { and } f(1)=3\). Let \(N=2\). Sketch two possible graphs of \(f\), one showing that \(f\) might not satisfy the conclusion of the Intermediate Value Theorem and one showing that \(f\) might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn't satisfy the hypothesis). Equation Transcription: Text Transcription: f [0,1] 0.25 f(0)=1 and f(1)=3 N=2 f f f

If \(f(x)=x^{2}+10 \sin x\) show that there is a number \(c\) such that \(f(c)=1000\). Equation Transcription: Text Transcription: f(x)=x^2+10 sin x c f(c)=1000

Suppose \(f\) is continuous on \({[1,5]}\) and the only solutions of the equation \(f(x)=6\) are \(x=1 \text { and } x=4\). If \(f(2)=8\), explain why \(f(3)>6\) Equation Transcription: Text Transcription: f [1,5] f(x)=6 x=1 and x=4 f(2)=8 f(3)>6

Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval. \(-x^{3}+4 x+1=0,(-1,0)\) Equation Transcription: Text Transcription: -x^{3}+4 x+1=0,(-1,0)

Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval. \(\ln x=x-\sqrt{x,}(2,3)\) Equation Transcription: Text Transcription: \ln x=x-\sqrt{x,}(2,3)

Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval. \(e^{x}=3-2 x,(0,1)\) Equation Transcription: Text Transcription: e^{x}=3-2 x,(0,1)

Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval. \(\sin x=x^{2}-x,(1,2)\) Equation Transcription: Text Transcription: \sin x=x^{2}-x,(1,2)

(a) Prove that the equation has at least one real solution. (b) Use a calculator to find an interval of length 0.01 that contains a solution. \(\cos x=x^{3}\) Equation Transcription: Text Transcription: \cos x=x^{3}

(a) Prove that the equation has at least one real solution. (b) Use a calculator to find an interval of length 0.01 that contains a solution. ln x = 3 - 2x

(a) Prove that the equation has at least one real solution. (b) Find the solution correct to three decimal places, by graphing. \(100 e^{-x / 100}=0.01 x^{2}\)

(a) Prove that the equation has at least one real solution. (b) Find the solution correct to three decimal places, by graphing. \(\arctan x=1-x\) Equation Transcription: Text Transcription: \arctan x=1-x

Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. \(y=\sin x^{3},(1,2)\) Equation Transcription: Text Transcription: y=\sin x^{3},(1,2)

Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. \(y=x^{2}-3+1 / x\) Equation Transcription: Text Transcription: y=x^{2}-3+1 / x

Prove that f is continuous at a if and only if \(\lim _{h \rightarrow 0} f(a+h)=f(a)\) Equation Transcription: Text Transcription: \lim _{h \rightarrow 0} f(a+h)=f(a)

To prove that sine is continuous, we need to show that \(\lim _{x \rightarrow a} \sin x=\sin a\) for every real number a. By Exercise 65 an equivalent statement is that \(\lim _{h \rightarrow 0} \sin (a+h)=\sin a\) Use (6) to show that this is true.

Prove that cosine is a continuous function.

(a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5.

Use Theorem 8 to prove Limit Laws 6 and 7 from Section 2.3.

Is there a number that is exactly 1 more than its cube?

For what values of x is f continuous? \(f(x)= \begin{cases}0 & \text { if } x \text { is rational } \\ 1 & \text { if } x \text { is irrational }\end{cases}\) Equation Transcription: { Text Transcription: f(x)= \begin{cases}0 & \text { if } x \text { is rational } \\ 1 & \text { if } x \text { is irrational

For what values of x is t continuous? \(g(x)= \begin{cases}0 & \text { if } x \text { is rational } \\ x & \text { if } x \text { is irrational }\end{cases}\) Equation Transcription: { Text Transcription: g(x)= \begin{cases}0 & \text { if } x \text { is rational } \\ x & \text { if } x \text { is irrational

Show that the function \(f(x)=\left\{\begin{array}{ll}x^{4} \sin (1 / x) & \text { if } x \neq 0 \\0 & \text { if } x=0\end{array}\right.\) is continuous on \((-\infty, \infty)\)

If a and b are positive numbers, prove that the equation \(\frac{a}{x^{3}+2 x^{2}-1}+\frac{b}{x^{3}+x-2}=0 \) has at least one solution in the interval \((-1,1)\). Equation Transcription: Text Transcription: \frac{a}{x^{3}+2 x^{2}-1}+\frac{b}{x^{3}+x-2}=0 (-1,1)

A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

Absolute Value and Continuity (a) Show that the absolute value function \(F(x)=|x| \) is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is \(|f|\) (c) Is the converse of the statement in part (b) also true? In other words, if \(|f|\) is continuous, does it follow that \(f\) is continuous? If so, prove it. If not, find a counterexample. Equation Transcription: Text Transcription: F(x)=| x | | f | | f | f