MATH 1830 Notes

Unit 2 Applications of Derivatives

2.2 First Derivative Test


First Derivative Test

Analyze $f'(x)$ to Identify Intervals of Increase/Decrease and Extrema on the Graph of $f(x).$

  1. Find $f'(x)$
  2. Identify all critical numbers and partitions for the function.
    1. Values of $x$ where$f'(x)=0$ are critical numbers.
    2. Values of $x$ where $f'(x)$ is undefined are partitions.
    3. Values of $x$ where $f(x)$ is undefined are partitions.
  3. Graph the critical numbers and partitions on a number line, separating the number line into intervals.
  4. Determine the intervals on which $f(x)$ is increasing /decreasing
    1. Test one point contained in the interval (do not use the end points of the interval).
    2. $f'(x)<0$ then the function $f(x)$ is DECREASING on the interval
    3. $f'(x)>0$ then the function $f(x)$ is INCREASING on the interval
  5. Identify local maxima and minima of $f(x)$ using the First Derivative Test.
    1. On the interval $(a,c)$, a local maximum occurs at $f(b)$ when $f(x)$ is increasing for all $x$ in the interval $(a,b]$ and $f(x)$ is decreasing for all $x$ in the interval $[b,c)$.
    2. On the interval $(a,c)$, a local minimum occurs at $f(b)$ when $f(x)$ is decreasing for all $x$ in the interval $(a,b]$ and $f(x)$ is increasing for all $x$ in the interval $[b,c)$.

2.2A Video

Use the First Derivative Test to analyze the function. Identify x- & y-intercepts, and any holes or asymptotes (if they exist).

  1. $f(x)={{x}^{3}}+6{{x}^{2}}+9x$

    1. x-intercept(s)

      ${{x}^{3}}+6{{x}^{2}}+9x=0$

      $x\left( {{x}^{2}}+6x+9 \right)=0 $

      $x{\left( x+3 \right)}{\left( x+3 \right)}=0 $

      $x=0\,\,,\,\,x=-3$

      $\left( 0,0 \right)\,\text{and}\,\left( -3,0 \right)$

    2. y-intercept

      $f\left( 0 \right)={{\left( 0 \right)}^{3}}+6{{\left( 0 \right)}^{2}}+9\left( 0 \right)=0$

      $\left( 0,0 \right)$

    3. Holes & Asymptotes

      There are no holes or asymptotes. The function is a polynomial.

    2.  

    2.2B Video

    Use the First Derivative Test to analyze the function. Identify intervals of increase, decrease, and extrema on the graph of the function.

  2. $f(x)={{x}^{3}}+6{{x}^{2}}+9x$

    1. Increasing and Decreasing

      ${f}'(x)=3{{x}^{2}}+12x+9$

      Values of x where ${f}'(x)=0: \quad 3{{x}^{2}}+12x+9=0$

      $3\left( x+3 \right)\left( x+1 \right)=0$

      $3 \neq 0$

      $x=-3$

      $x=-1$

      Values of x where ${f}'\left( x \right)$ is undefined:

      There are no values of x where ${f}'\left( x \right)$ is undefined.

      Values of x where $f\left( x \right)$ is undefined:

      There are no values of x where $f\left( x \right)$ is undefined.

      Separate into intervals using: $x=-3$ and $x=-1$.

      Image of graph of F prime of x as it relates to the x axis. Separates graph into intervals divided by partitions where F prime equals zero, or is undefined. Also indicates intervals where derivative is positive and intervals where derivative is negative.
      Sign graph of ${f}'(x)$ reading left to right: positive, ${f}'\left( -3 \right)=0$, negative, ${f}'\left( -1 \right)=0$, positive

      Increasing:

      The graph of $f(x)$ is increasing on the intervals $\left( -\infty ,-3 \right)\cup \left( -1,\infty \right)$.

      Decreasing:

      The graph of $f(x)$ is decreasing on the interval $\left( -3,-1 \right)$.

    2. Local Maxima:

      $f\left( -3 \right)={{\left( -3 \right)}^{3}}+6{{\left( -3 \right)}^{2}}+9\left( -3 \right)=0$

      There is a local maximum at the point $\left( -3,0 \right)$.

      Local Minima:

      $f\left( -1 \right)={{\left( -1 \right)}^{3}}+6{{\left( -1 \right)}^{2}}+9\left( -1 \right)=-4$

      There is a local minimum at the point $\left( -1,-4 \right)$.

      blank 4 quadrant coordinate plane

      Graph of F of x indicating intervals of increase, intervals of decrease, local maxima and local minima

    3.  

    2.2 A&B Lecture

    Use the First Derivative Test to analyze the function. Identify x- & y-intercepts, any holes or asymptotes (if they exist), intervals of increase/decrease, and extrema on the graph of the function.

  3. $f(x)=3{{x}^{2}}+5x-2$

    1. x intercept(s):

      $ 0=3{{x}^{2}}+5x-2 $

      $ 0=\left( 3x-1 \right)\left( x+2 \right) $

      $x=\frac{1}{3}\,\,,\,\,x=-2$

      $\left( \frac{1}{3},0 \right)\,\text{and}\,\left( -2,0 \right)$

    2. y intercept:

      $f\left( 0 \right)=0+0-2=-2$

      $\left( 0,-2 \right)$

    3. Holes & Asymptotes:

      There are no holes or asymptotes. The function is a polynomial.

    4. Increasing and Decreasing

      ${f}'(x)=6x+5$

      Values of x where ${f}'(x)=0:$

      $6x+5=0$

      $6x=-5$

      $x=-\frac{5}{6}$

      Values of x where ${f}''\left( x \right)\;$is undefined:

      There are no values of x where ${f}''\left( x \right)\;$is undefined.

      Values of x where $f\left( x \right)\;$is undefined:

      There are no values of x where $f\left( x \right)\;$is undefined.

      Separate into intervals using: $x=-\frac{5}{6}$.

      Image of graph of F prime of x as it relates to the x axis. Separates graph into intervals divided by partitions where F prime equals zero, or is undefined. Also indicates intervals where derivative is positive and intervals where derivative is negative.
      Sign graph of ${f}'(x)$ reading left to right: negative, ${f}'\left( -\frac{5}{6} \right)=0$ , positive

      Increasing:

      The graph of $f(x)$ is increasing on the interval $\left( -\frac{5}{6},\infty \right)$.

      Decreasing:

      The graph of $f(x)$ is decreasing on the interval $\left( -\infty ,-\frac{5}{6} \right)$.

    5. local max:

      There are no local maxima.

      local min:

      $f\left( -\frac{5}{6} \right)=3{{\left( -\frac{5}{6} \right)}^{2}}+5\left( -\frac{5}{6} \right)-2\approx 4.08$

      There is a local minimum at the point $\left( -\frac{5}{6},-4.08 \right)$.

      Graph of F of x indicating intervals of increase, intervals of decrease, local maxima and local minima

    4.  

    2.2 Group Work

  4. Use the First Derivative Test to analyze $f'(x)$ and identify intervals of increase/decrease and extrema on the graph of $f(x)$.

    $f(x)=2x^3+3x^2-12x+5$

    1. Increasing and Decreasing

      ${f}'(x)=6x^2-6x-12$

      Values of x where ${f}'(x)=0:$

      $6x^2-6x-12=0$

      $6(x^2+x-2)=0$

      $6(x+2)(x-1)=0$

      $x=-2 & 1$

      Values of x where ${f}'\left( x \right)\;$is undefined:

      There are no values of x where ${f}'\left( x \right)\;$is undefined.

      Values of x where $f\left( x \right)\;$is undefined:

      There are no values of x where $f\left( x \right)\;$is undefined.

      Separate into intervals using: $x=-2 & 1.

      Image of graph of number line with -2 and 1 marked with dotted lines above to divide the number line into 3 regions. -3, 0, and 2 are labeled under the number line as test points. There is an arrow point up in the regions from negative infinity to -2, an arrow pointing down on the interval from -2 to 1, and another arrow pointing up on the interval from 1 to infinity.

      Increasing:

      The graph of $f(x)$ is increasing on the interval $(-\infty,-2) ∪ (1, \infty) $.

      Decreasing:

      The graph of $f(x)$ is decreasing on the interval $(-2,1)$.

    2. local max:

      $f(-2)=2(-2)^3+3(-2)^2-12(-2)+5$

      There is local maxima at (-2, 25).

      local min:

      $f(1)=2(1)^3+3(1)^2-12(1)+5$

      There is a local minimum at the point $(1, -2)$.

  5. The annual first quarter change in revenue for Apple, Inc. is given by the regression model: $$f(x)=-0.005x^4+0.113x^3-0.889x^2+7.946x-5.346$$ where x is Years Since 1998 until 2016.

    Use the First Derivative Test to analyze $f'(x)$ and identify intervals of increase/decrease and extrema on the graph of $f(x)$.

    $f'(x)=-0.02x^3+0.339x^2-1.778x+7.946$

    1. Identify the values of x where $f'(x)=0$:

      $f'(x)=0$ at $x=12.358$

      Values of x where $f'(x)$ is undefined:

      There are no values of x where $f'(x)$ is undefined.

      Values of x where $f(x)$ is undefined:

      There are no values of x where $f(x)$ is undefined.

      Separate the graph of $f'(x)$ into intervals using: $x=12.358$.

      graph of first derivative
      Sign graph of ${f}'(x)$ reading left to right: positive, ${f}'\left( 12.358 \right)=0$, negative

      Intervals of Increase and Intervals of Decrease

      Increasing:

      $[0,12.358)$

      The percent change in revenue was increasing from 1998 into 2010.

      Decreasing:

      $(12.358,18]$

      The percent change in revenue was decreasing in 2010 through 2016.

    2. Extrema

      Local maximum:

      $(12.358, 53.732)$

      From the model, the maximum percent change in revenue occurs in 2010 (at x=12.358 years after 1998).

      Local minimum:

      There are no local minima.

    6.  

    2.2 Additional Practice

    Use the First Derivative Test to analyze the function. Identify intervals of increase/decrease and extrema on the graph of the function.

  6. Using data from the Federal Reserve, the S&P 500 annual percent return on investments for the years 2008-2014 can be modeled by the following equation: $$A(t) = - 1.64{t^4} + 20.85{t^3} - 86.05{t^2} + 127.87t - 36.24$$ where t is in years since 2008 and A(t) is in percent.

    Source: http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html

    Use the First Derivative Test to analyze $A'(t)$ and identify intervals of increase/decrease and extrema on the graph of $A(t)$.

    ${A}'\left( t \right)=-6.56{{t}^{3}}+62.55{{t}^{2}}-172.10t+127.87$

    1. Graph the critical numbers on a number line and determine the sign for each interval.

      Values of t where ${A}'\left( t \right)=0$:

      $0=-6.56{{t}^{3}}+62.55{{t}^{2}}-172.10t+127.87$

      ${A}'\left( t \right)=0$ when $t\approx 1.20$, $t\approx 3.10$, $t\approx 5.23$

      Values of t where ${A}'\left( t \right)$ is undefined:

      There are no values of t where ${A}'\left( t \right)$ is undefined.

      Values of t where ${A}\left( t \right)$ is undefined:

      There are no values of t where ${A}\left( t \right)$ is undefined.

      Separate into intervals using: $t=1.20$, $t=3.10$ and $t=5.23$.

      Graph of the first derivative of the function A of t
      Sign graph of ${A}'(x)$ from left: positive, ${A}'\left( 1.20 \right)\approx 0$, negative, ${A}'\left( 3.10 \right)\approx 0$, positive, ${A}'\left( 5.23 \right)\approx 0$, negative

      Determine the intervals on which $A(t)$ is increasing/decreasing.

      The function is increasing on the intervals: $\left( 0,1.20 \right)\cup \left( 3.10,5.23 \right).$

      APR increased from 2008 into 2009 and from 2011 into 2013.

      The function is decreasing on the intervals: $\left( 1.20,3.10 \right)\cup \left( 5.23,6 \right).$

      APR decreased from 2009 into 2011 and in 2013.

    2. Identify local maxima and minima for $A(t)$.

      There are two Local Maxima at the points: $\left( 1.20,25.92 \right)$ and $\left( 5.23,34.50 \right).$

      There is a Local Minimum at the point: $\left( 3.10,2.90 \right).$

  7. Using data from Statista, the total annual amount spent on the purchase of golf equipment in the United States for the years 2008-2014 can be modeled by the following equation:

    $$A(t)=-31.94{{t}^{3}}+301.16{{t}^{2}}-665.61t+3454.63$$ where t is in years since 2008 and A(t) is in millions of dollars.

    Source: href="https://www.statista.com/statistics/201038/purchases-of-golf-equipment-in-the-us-since-2007/">http://www.statista.com/statistics/201038/purchases-of-golf-equipment-in-the-us-since-2007/

    Use the First Derivative Test to analyze the function. Identify intervals of increase/decrease and extrema on the graph of the function.

    ${A}'\left( t \right)=-95.82{{t}^{2}}+602.32t-665.61$

    1. Graph the critical numbers on a number line and determine the sign for each interval.

      Values of t where ${A}'\left( t \right)=0:$

      $t=1.43$ and $t=4.86$

      Values of t where ${A}'\left( t \right)$ is undefined:

      There are no values of t where ${A}'\left( t \right)$is undefined.

      Values of t where ${A}\left( t \right)$ is undefined:

      There are no values of t where $A\left( t \right)$ is undefined.

      Separate into intervals using: $t=1.43$ and $t=4.86$.

      Graph of the first derivative of the function A of t
      Sign chart for ${A}'\left( t \right)$ reading left to right: negative, ${A}'\left( 1.43 \right)\approx 0$, positive, ${A}'\left( 4.86 \right)\approx 0$, negative

      Determine the intervals on which $A(t)$ is increasing/decreasing.

      The function is increasing on the interval: $\left( 1.43,4.86 \right).$

      From 2009 to 2012, purchase amounts of golf equipment in the US increased each year.

      The function is decreasing on the intervals: $\left( 0,1.43 \right)\cup \left( 4.86,6 \right).$

      In 2008 into 2009 and in 2013 and 2014, purchase amounts of golf equipment decreased.

    2. Identify local maxima and minima for $A(t)$.

      There is a Local Maximum at the point: $\left( 4.86,3666.61 \right).$

      There is a Local Minimum at the point: $\left( 1.43,3025.25 \right).$