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Mathematics LibreTexts

12.R: Introduction to Calculus (Review)

  • Page ID
    19009
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    12.1: Finding Limits - Numerical and Graphical Approaches

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

    R 12.1.1.png

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

    Answer

    \(2\)

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

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

    Answer

    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?

    Answer

    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
    Answer

    \(3\)

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

    Answer

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

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

    Answer

    \(2\)

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

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

    Answer

    \(-15\)

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

    Answer

    \(3\)

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

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

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

    Answer

    \(12\)

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

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

    Answer

    \(-10\)

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

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

    Answer

    \(-\dfrac{1}{9}\)

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

    Answer

    At \(x=4\), the function has a vertical asymptote.

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

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

    Answer

    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) \(f(x)=x^2-2x-15\)

    Answer

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

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

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

    Answer

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

    Answer

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

    Answer

    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) \(f(x)=5\)

    Answer

    \(0\)

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

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

    Answer

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

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

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

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

    Answer

    \(4\)

    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. \[f(x)=-x^3+4x;\; x=2 \nonumber \]

    Answer

    \(y=-8x+16\)

    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