Chapter 2

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=x^{3}+1\) (a) $$[2,3] \quad$$ \((b)\lceil- 1,1]\)

In Exercises \(1 - 4 ,\) an object dropped from rest from the top of a tall building falls \(y = 16 t ^ { 2 }\) feet in the first \(t\) seconds. Find the average speed during the first 3 seconds of fall.

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\frac{1}{(x+2)^{2}}$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\cos \left(\frac{1}{x}\right)$$

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=\sqrt{4 x+1}\) (a) $$[0,2] \quad$$ (b) $$[10,12]$$

In Exercises \(1 - 4 ,\) an object dropped from rest from the top of a tall building falls \(y = 16 t ^ { 2 }\) feet in the first \(t\) seconds. Find the average speed during the first 4 seconds of fall.

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\frac{x+1}{x^{2}-4 x+3}$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{\sin 2 x}{x}$$

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=e^{x}\) (a) $$[-2,0] \quad$$ (b) $$[1,3]$$

In Exercises \(1 - 4 ,\) an object dropped from rest from the top of a tall building falls \(y = 16 t ^ { 2 }\) feet in the first \(t\) seconds. Find the speed of the object at \(t = 3\) seconds and confirm your answer algebraically.

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\frac{1}{x^{2}+1}$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{e^{-x}}{x}$$

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=\ln x\) (a) \([1,4], \quad\) ( b) \([100,103]\)

In Exercises \(1 - 4 ,\) an object dropped from rest from the top of a tall building falls \(y = 16 t ^ { 2 }\) feet in the first \(t\) seconds. Find the speed of the object at \(t = 4\) seconds and confirm your answer algebraically.

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=|x-1|$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{3 x^{3}-x+1}{x+3}$$

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=\cot t\) (a) \([\pi / 4,3 \pi / 4] \quad\) (b) \([\pi / 6, \pi / 2]\)

In Exercises 5 and \(6 ,\) use \(\lim _ { x \rightarrow c } k = k , \lim _ { x \rightarrow c } x = c ,\) and the properties of limits to find the limit. $$\lim _ { x \rightarrow c } \left( 2 x ^ { 3 } - 3 x ^ { 2 } + x - 1 \right)$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\sqrt{2 x+3}$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{3 x+1}{|x|+2}$$

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=2+\cos t\) (a) \([0, \pi]\) \((\mathbf{b})[-\pi, \pi]\)

In Exercises 5 and \(6 ,\) use \(\lim _ { x \rightarrow c } k = k , \lim _ { x \rightarrow c } x = c ,\) and the properties of limits to find the limit. $$\lim _ { x \rightarrow c } \frac { x ^ { 4 } - x ^ { 3 } + 1 } { x ^ { 2 } + 9 }$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\sqrt[3]{2 x-1}$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{2 x-1}{|x|-3}$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 1 / 2 } 3 x ^ { 2 } ( 2 x - 1 )$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=|x| / x$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{x}{|x|}$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 4 } ( x + 3 ) ^ { 1998 }$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\cot x$$

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{|x|}{|x|+1}$$

In Exercises \(9-12,\) at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the tangent. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window. $$v=x^{2} \quad\( at \)\quad x=-2$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow 1 } \left( x ^ { 3 } + 3 x ^ { 2 } - 2 x - 17 \right)$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=e^{1 / x}$$

In Exercises 9-12, find the limit and confirm your answer using the Sandwich Theorem. $$\lim _{x \rightarrow \infty} \frac{1-\cos x}{x^{2}}$$

In Exercises \(9-12,\) at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the tangent. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window. $$y=x^{2}-4 x\( at \)x=1$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { y \rightarrow 2 } \frac { y ^ { 2 } + 5 y + 6 } { y + 2 }$$

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\ln (x+1)$$

In Exercises 9-12, find the limit and confirm your answer using the Sandwich Theorem. $$\lim _{x \rightarrow-\infty} \frac{1-\cos x}{x^{2}}$$

In Exercises \(9-12,\) at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the tangent. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window. $$y=\frac{1}{x-1}\( at \)x=2$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { y \rightarrow - 3 } \frac { y ^ { 2 } + 4 y + 3 } { y ^ { 2 } - 3 }$$

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ (a) Does \(f(-1)\) exist? (b) Does \(\lim _{x \rightarrow-1^{+}} f(x)\) exist? (c) Does \(\lim _{x \rightarrow-1^{+}} f(x)=f(-1) ?\) (d) Is \(f\) continuous at \(x=-1 ?\)

In Exercises 9-12, find the limit and confirm your answer using the Sandwich Theorem. $$\lim _{x \rightarrow-\infty} \frac{\sin x}{x}$$

In Exercises \(9-12,\) at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the tangent. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window. $$y=x^{2}-3 x-1 \quad\( at \)\quad x=0$$

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ (a) Does \(f(1)\) exist? (b) Does \(\lim _{x \rightarrow 1} f(x)\) exist? (c) Does \(\lim _{x \rightarrow 1} f(x)=f(1) ?\) (d) Is \(f\) continuous at \(x=1 ?\)

In Exercises 13 and \(14,\) find the slope of the curve at the indicated point. $$f(x)=|x| \quad\( at \)\quad\( (a) \)x=2 \quad\( (b) \)x=-3$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 2 } ( x - 6 ) ^ { 2 / 3 }$$

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ (a) Is \(f\) defined at \(x=2 ?\) (Look at the definition of \(f . )\) (b) Is \(f\) continuous at \(x=2 ?\)

In Exercises 13-20, use graphs and tables to find the limits. $$\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}$$

In Exercises 13 and \(14,\) find the slope of the curve at the indicated point. $$f(x)=|x-2|\( at \)x=1$$

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow 2 } \sqrt { x + 3 }$$