# 7.R: Introduction to Calculus (Review)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## 12.1: Finding Limits - Numerical and Graphical Approaches

For the exercises 1-6, use the Figure below.

1) $$\lim \limits_{x \to -1^+}f(x)$$

$$2$$

2) $$\lim \limits_{x \to -1^-}f(x)$$

3) $$\lim \limits_{x \to -1}f(x)$$

does not exist

4) $$\lim \limits_{x \to 3}f(x)$$

5) At what values of $$x$$ is the function discontinuous? What condition of continuity is violated?

Discontinuous at $$x=-1\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$$, $$x=3\left (\text{ jump discontinuity} \right )$$, and $$x=7\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$$.

6) Using the Table below, estimate $$\lim \limits_{x \to 0}f(x)$$.

$$x$$ $$F(x)$$
−0.1 2.875
−0.01 2.92
−0.001 2.998
0 Undefined
0.001 2.9987
0.01 2.865
0.1 2.78145
0.15 2.678

$$3$$

For the exercises 7-9, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $$x$$ approaches $$a$$. If the function has limit as $$x$$ approaches $$a$$, state it. If not, discuss why there is no limit.

7) $$f(x)=\begin{cases} \left | x \right |-1 & \text{ if } x\neq 1 \\ x^3 & \text{ if } x= 1 \end{cases} a=1$$

8) $$f(x)=\begin{cases} \dfrac{1}{x+1} & \text{ if } x= -2 \\ (x+1)^2 & \text{ if } x\neq -2 \end{cases} a=-2$$

$$\lim \limits_{x \to -2}f(x)=1$$

9) $$f(x)=\begin{cases} \sqrt{x+3} & \text{ if } x<1 \\ -\sqrt[3]{x} & \text{ if } x>1 \end{cases} a=1$$

## 12.2: Finding Limits - Properties of Limits

For the exercises 1-6, find the limits if $$\lim \limits_{x \to c} f(x)=-3$$ and $$\lim \limits_{x \to c} g(x)=5$$.

1) $$\lim \limits_{x \to c} (f(x)+g(x))$$

$$2$$

2) $$\lim \limits_{x \to c} \dfrac{f(x)}{g(x)}$$

3) $$\underset{x \to c}{\lim } (f(x)\cdot g(x))$$

$$-15$$

4) $$\lim \limits_{x \to 0^+} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$$

5) $$\lim \limits_{x \to 0^-} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$$

$$3$$

6) $$\lim \limits_{x \to 3^+} (3x-〚x〛)$$

For the exercises 7-11, evaluate the limits using algebraic techniques.

7) $$\lim \limits_{h \to 0} \left ( \dfrac{(h+6)^2-36}{h} \right )$$

$$12$$

8) $$\lim \limits_{x \to 25} \left ( \dfrac{x^2-625}{\sqrt{x}-5} \right )$$

9) $$\lim \limits_{x \to 1} \left ( \dfrac{-x^2-9x}{x} \right )$$

$$-10$$

10) $$\lim \limits_{x \to 4} \left ( \dfrac{7-\sqrt{12x+1}}{x-4} \right )$$

11) $$\lim \limits_{x \to 3} \left ( \dfrac{\frac{1}{3}+\frac{1}{x}}{3+x} \right )$$

$$-\dfrac{1}{9}$$

## 12.3: Continuity

For the exercises 1-5, use numerical evidence to determine whether the limit exists at $$x=a$$. If not, describe the behavior of the graph of the function at $$x=a$$.

1) $$f(x)=\dfrac{-2}{x-4};\; a=4$$

2) $$f(x)=\dfrac{-2}{(x-4)^2};\; a=4$$

At $$x=4$$, the function has a vertical asymptote.

3) $$f(x)=\dfrac{-x}{x^2-x-6};\; a=3$$

4) $$f(x)=\dfrac{6x^2+23x+20}{4x^2-25};\; a=-\dfrac{5}{2}$$

removable discontinuity at $$a=-\dfrac{5}{2}$$

5) $$f(x)=\dfrac{\sqrt{x}-3}{9-x};\; a=9$$

For the exercises 6-12, determine where the given function $$f(x)$$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

6) $$f(x)=x^2-2x-15$$

continuous on $$(-\infty, \infty)$$

7) $$f(x)=\dfrac{x^2-2x-15}{x-5}$$

8) $$f(x)=\dfrac{x^2-2x}{x^2-4x+4}$$

removable discontinuity at $$x=2$$. $$f(2)$$ is not defined, but limits exist.

9) $$f(x)=\dfrac{x^3-125}{2x^2-12x+10}$$

10) $$f(x)=\dfrac{x^2-\frac{1}{x}}{2-x}$$

discontinuity at $$x=0$$ and $$x=2$$. Both $$f(0)$$ and $$f(2)$$ are not defined.

11) $$f(x)=\dfrac{x+2}{x^2-3x-10}$$

12) $$f(x)=\dfrac{x+2}{x^3+8}$$

removable discontinuity at $$x=-2$$. $$f(-2)$$ is not defined.

## 12.4: Derivatives

For the exercises 1-5, find the average rate of change $$f(x)=\dfrac{f(x+h)-f(x)}{h}$$.

1) $$f(x)=3x+2$$

2) $$f(x)=5$$

$$0$$

3) $$f(x)=\dfrac{1}{x+1}$$

4) $$f(x)=\ln (x)$$

$$f(x)=\dfrac{\ln (x+h)-\ln (x)}{h}$$

5) $$f(x)=e^{2x}$$

For the exercises 6-7, find the derivative of the function.

6) $$f(x)=4x-6$$

$$4$$

7) $$f(x)=5x^2-3x$$

8) Find the equation of the tangent line to the graph of $$f(x)$$ at the indicated $$x$$ value. $f(x)=-x^3+4x;\; x=2 \nonumber$

$$y=-8x+16$$

9) For the following exercise, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. $f(x)=\dfrac{x}{\left | x \right |} \nonumber$

10) Given that the volume of a right circular cone is $$V=\dfrac{1}{3}\pi r^2h$$ and that a given cone has a fixed height of $$9$$ cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is $$2$$ cm. Give an exact answer in terms of $$π$$.

$$12\pi$$

## Practice Test

For the exercises 1-6, use the graph of $$f$$ in the Figure below.

1) $$f(1)$$

$$3$$

2) $$\lim \limits_{x \to -1^+} f(x)$$

3) $$\lim \limits_{x \to -1^-} f(x)$$

$$0$$

4) $$\lim \limits_{x \to -1} f(x)$$

5) $$\lim \limits_{x \to -2} f(x)$$

$$-1$$

6) At what values of $$x$$ is $$f$$ discontinuous? What property of continuity is violated?

7) $$f(x)=\begin{cases} \dfrac{1}{3}-3 & \text{ if } x\leq 2 \\ x^3+1 & \text{ if } x>2 \end{cases} a=2$$

$$\lim \limits_{x \to 2^-} f(x)=-\dfrac{5}{2}a$$ and $$\lim \limits_{x \to 2^+} f(x)=9$$

Thus, the limit of the function as $$x$$ approaches $$2$$ does not exist.

8) $$f(x)=\begin{cases} x^3+1 & \text{ if } x<1 \\ 3x^2-1 & \text{ if } x=1\; a=1 \\ -\sqrt{x+3}+4 & \text{ if } x>1 \end{cases}$$

For the exercises 9-11, evaluate each limit using algebraic techniques.

9) $$\lim \limits_{x \to -5} \left ( \dfrac{\frac{1}{5}+\frac{1}{x}}{10+2x} \right )$$

$$-\dfrac{1}{50}$$

10) $$\lim \limits_{h \to 0} \left ( \dfrac{\sqrt{h^2+25}-5}{h^2} \right )$$

11) $$\lim \limits_{h \to 0} \left ( \dfrac{1}{h}-\dfrac{1}{h^2+h} \right )$$

$$1$$

For the exercises 12-13, determine whether or not the given function $$f$$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12) $$f(x)=\sqrt{x^2-4}$$

13) $$f(x)=\dfrac{x^3-4x^2-9x+36}{x^3-3x^2+2x-6}$$

removable discontinuity at $$x=3$$

For the exercises 14-16, use the definition of a derivative to find the derivative of the given function at $$x=a$$.

14) $$f(x)=\dfrac{3}{5+2x}$$

15) $$f(x)=\dfrac{3}{\sqrt{x}}$$

$$f'(x)=-\dfrac{3}{2a^{\frac{3}{2}}}$$

16) $$f(x)=2x^2+9x$$

17) For the graph in the Figure below, determine where the function is continuous/discontinuous and differentiable/not differentiable.

discontinuous at $$-2,0$$, not differentiable at $$-2,0, 2$$.

For the exercises 18-19, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

18) $$f(x)=\left | x-2 \right | - \left | x+2 \right |$$

19) $$f(x)=\dfrac{2}{1+e^{\frac{2}{x}}}$$

not differentiable at $$x=0$$ (no limit)

For the exercises 20-24, explain the notation in words when the height of a projectile in feet, $$s$$, is a function of time $$t$$ in seconds after launch and is given by the function $$s(t)$$.

20) $$s(0)$$

21) $$s(2)$$

the height of the projectile at $$t=2$$ seconds

22) $$s'(2)$$

23) $$\dfrac{s(2)-s(1)}{2-1}$$

the average velocity from $$t=1$$ to $$t=2$$

24) $$s(t)=0$$

For the exercises 25-28, use technology to evaluate the limit.

25) $$\lim \limits_{x \to 0}\dfrac{\sin (x)}{3x}$$

$$\dfrac{1}{3}$$

26) $$\lim \limits_{x \to 0}\dfrac{\tan ^2(x)}{2x}$$

27) $$\lim \limits_{x \to 0}\dfrac{\sin (x)(1-\cos (x))}{2x^2}$$

$$0$$

28) Evaluate the limit by hand.

$\lim \limits_{x \to 1}f(x), \text{ where } f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$

At what value(s) of $$x$$ is the function below discontinuous?

$f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$

For the exercises 29-32, consider the function whose graph appears in Figure.

29) Find the average rate of change of the function from $$x=1$$ to $$x=3$$.

$$2$$

30) Find all values of $$x$$ at which $$f'(x)=0$$.

$$x=1$$

31) Find all values of $$x$$ at which $$f'(x)$$ does not exist.

32) Find an equation of the tangent line to the graph of $$f$$ the indicated point: $$f(x)=3x^2-2x-6,\; x=-2$$

$$y=-14x-18$$

For the exercises 33-34, use the function $$f(x)=x(1-x)^{\frac{2}{5}}$$

33) Graph the function $$f(x)=x(1-x)^{\tfrac{2}{5}}$$ by entering $$f(x)=x\left ((1-x)^2 \right )^{\tfrac{1}{5}}$$ and then by entering $$f(x)=x\left ((1-x)^{\tfrac{1}{5}} \right )^2$$.

34) Explore the behavior of the graph of $$f(x)$$ around $$x=1$$ by graphing the function on the following domains, $$[0.9, 1.1], [0.99, 1.01], [0.999, 1.001]$$, and $$[0.9999, 1.0001]$$. Use this information to determine whether the function appears to be differentiable at $$x=1$$.

The graph is not differentiable at $$x=1$$ (cusp).

For the exercises 35-42, find the derivative of each of the functions using the definition: $$\lim \limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$$

35) $$f(x)=2x-8$$

36) $$f(x)=4x^2-7$$

$$f'(x)=8x$$

37) $$f(x)=x-\dfrac{1}{2}x^2$$

38) $$f(x)=\dfrac{1}{x+2}$$

$$f'(x)=-\dfrac{1}{(2+x)^2}$$

39) $$f(x)=\dfrac{3}{x-1}$$

40) $$f(x)=-x^3+1$$

$$f'(x)=-3x^2$$

41) $$f(x)=x^2+x^3$$

42) $$f(x)=\sqrt{x-1}$$

$$f'(x)=-\dfrac{1}{2\sqrt{x-1}}$$