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# 12.R: Introduction to Calculus (Review)

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## 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