Skip to main content

Workbook for Stewart’s "Calculus"

Cory Wilson

Section 3.3 How Derivatives Affect the Shape of a Graph

Subsection 3.3.1 Before Class

https://mymedia.ou.edu/media/3.3-1.mp4/1_6ntbtqd6
Figure 23. Pre-Class Video 1
https://mymedia.ou.edu/media/3.3-2/1_q3uahwvk
Figure 24. Pre-Class Video 2

Subsubsection 3.3.1.1 Increasing/Decereasing Test

Example 3.3.2.
Let \(f(x) = (x-5)^2 + 1\text{.}\)
  1. Sketch the graph of \(f\text{,}\) and visually determine where the function is increasing and decreasing.
  2. Now, use the Increasing/Decreasing Test to determine the intervals where \(f\) is increasing and decreasing.
Solution.
  1. The graph of \(f(x)=(x-5)^2+1\) on the interval \([3,7]\text{.}\)
    The function is increasing on \((5,\infty)\) and decreasing on \((-\infty,5)\text{.}\)
  2. Using the Increasing/Decreasing Test, we again find that the function is increasing on \((5,\infty)\) and decreasing on \((-\infty,5)\text{.}\)
Example 3.3.3.
Let \(f(x) = 3x^4-4x^3-12x^2+5\text{.}\) Find the intervals where \(f\) is increasing and decreasing.
Solution.
\(f(x)\) is increasing on \((-1,0)\cup (2,\infty)\) and decreasing on \((-\infty,-1)\cup (0,2)\)

Subsubsection 3.3.1.2 Local Extrema

Fermat’s Theorem tells us that if we have a local max or min at input \(x = c\text{,}\) then \(c\) is a critical number of \(f\text{.}\) We need some more machinery in order to classify extrema.
Example 3.3.5.
Find the local maximum and minimum values of the function \(g(x) = x + 2\sin x\) on \(0\leq x\leq 2\pi\text{.}\)
Solution.
The local max occurs at \(x=\dfrac{2\pi}{3}\) and the local min occurs at \(x=\dfrac{4\pi}{3}\)
Example 3.3.6.
Let \(f(x) = 200 + 8x^3 + x^4\text{.}\) Find the intervals where \(f\) is increasing, decreasing, and identify local extrema of \(f\text{.}\)
Solution.
The function is increasing on \((-6,0)\cup (0,\infty)\) and decreasing on \((-\infty,-6)\text{.}\) There is a local minmum at \((-6,-232)\text{.}\)

Subsubsection 3.3.1.3 Concavity

The first derivative tells us information about where a function is increasing or decreasing, and where it has horizontal tangent lines. The second derivative tells us information about how a function bends.
Definition 3.3.7. Concave Up/Concave Down.
If the graph of \(f\) lies above all of its tangents on an interval \(I\text{,}\) then it is said to be concave up on \(I\text{;}\) if \(f\) lies below all of its tangents on \(I\text{,}\) then it is said to be concave down on \(I\text{.}\)
Definition 3.3.8. Inflection Point.
A point \((c,f(c))\) on a curve \(y = f(x)\) is called an inflection point if \(f\) is continuous there and the curve changes concavity.
Example 3.3.9.
Determine the sign of the second derivative for the function graphed below, and note any inflection points.
The graph of \(f(x)=x^3\) on the interval \([-2,2]\text{.}\)
Solution.
\(f''(x) \lt 0 \) on \((-\infty,0)\) and \(f''(x) \gt 0\) on \((0,\infty)\text{.}\) There is an inflection point at \(x=0\text{.}\)

Subsection 3.3.2 Pre-Class Activities

Example 3.3.10.

Let \(f(x) = x^3-3x^2-9x+4\text{.}\) Find the intervals where \(f\) is increasing and decreasing.
Solution.
\(f(x)\) is increasing on \((-\infty,-1)\cup (3,\infty)\) and decreasing on \((-1,3)\text{.}\)

Example 3.3.11.

Given the picture below, find the open intervals where \(f\) is increasing, decreasing, concave up, concave down. Identify the coordinates of any inflection points.
Solution.
The function is increasing on \((1,3)\cup (4,6)\text{,}\) decreasing on \((0,1)\cup (3,4)\text{,}\) concave up on \((0,2)\text{,}\) and concave down on \((2,4)\cup (4,6)\text{.}\) There is an inflection point at \((2,3)\text{.}\)

Example 3.3.12.

The graph of the derivative, \(f'\text{,}\) is given below. On what interval(s) is \(f\) increasing or decreasing? At what inputs does \(f\) have a local maximum or minimum?
Solution.
\(f(x)\) is increasing on \((1,5)\) and decreasing on \((0,1)\cup (5,6)\text{.}\) There is a local max at \(x=5\) and a local min at \(x=1\text{.}\)

Subsection 3.3.3 In Class

Example 3.3.14.

Let \(f(x) = x^3 - 3x\text{.}\)
  1. Find the intervals where \(f(x)\) is increasing and decreasing.
  2. Identify the local maxima and minima of \(f(x)\text{.}\)
  3. Where is \(f\) concave up? Where is it concave down?
  4. Identify any inflection points of \(f\text{.}\)
Solution.
  1. \(f(x)\) is increasing on \((-\infty,-1)\cup (1,\infty)\) and decreasing on \((-1,1)\)
  2. The max occurs at \((-1,2)\) and the min occurs at \((1,-2)\)
  3. \(f(x)\) is concave up on \((0,\infty)\) and concave down on \((-\infty,0)\)
  4. The inflection point is at \((0,0)\)

Example 3.3.15.

Sketch a possible graph for a function \(f\) that satisfies the conditions:
  1. \(f(0) = 0\text{,}\) \(f(2) = 3\text{,}\) \(f(4) = 6\text{,}\) \(f'(0) = f'(4) = 0\)
  2. \(f'(x) \gt 0\) for \(0 \lt x \lt 4\text{,}\) \(f'(x) \lt 0\) for \(x \lt 0\) and for \(x \gt 4\)
  3. \(f''(x) \gt 0\) for \(x \lt 2\text{,}\) \(f''(x) \lt 0\) for \(x \gt 2\)
Solution.
Answers vary

Example 3.3.17.

Sketch the curve \(y = x^4-4x^3\) by finding the intervals of increasing/decreasing, local maxima/minima, intervals of concavity, and inflection point(s).
Solution.
The graph of \(y = x^4-4x^3\) on the interval \([-2.5,5]\text{.}\)

Example 3.3.18.

Sketch the curve \(y=\dfrac{1}{2}x^4 -4x^2 + 3\)
Solution.
The graph of \(y=\dfrac{1}{2}x^4 -4x^2 + 3\) on the interval \([-3,3]\text{.}\)

Example 3.3.19.

Sketch the curve \(f(x)=x^3 - 12x + 2\)
Solution.
The graph of \(f(x)=x^3 - 12x + 2\) on the interval \([-4,4]\text{.}\)

Example 3.3.20.

Sketch the curve \(h(x) = 5x^3-3x^5\)
Solution.
The graph of \(h(x) = 5x^3-3x^5\) on the interval \([-1.5,1.5]\text{.}\)

Example 3.3.21.

Sketch the graph of a function that satisfies the criteria outlined below:
  1. Vertical asymptote at \(x = 0\text{.}\)
  2. \(f'(x) \gt 0\) if \(x \lt -2\text{,}\) \(f'(x) \lt 0\) if \(x \gt -2\) (for \(x\neq 0\)).
  3. \(f''(x) \lt 0\) if \(x \lt 0\text{,}\) \(f''(x) \gt 0\) if \(x \gt 0\text{.}\)
Solution.
Answers vary

Example 3.3.22.

Sketch the graph of a function that satisfies the criteria outlined below:
  1. \(f'(5) = 0\text{,}\) \(f'(x) \lt 0\) when \(x \lt 5\text{,}\) \(f'(x) \gt 0\) when \(x \gt 5\)
  2. \(f''(2) = 0\text{,}\) \(f''(8) = 0\text{,}\) \(f''(x) \lt 0 \) when \(x \lt 2\) or \(x \gt 8\text{,}\) \(f''(x) \gt 0\) for \(2\lt x\lt 8\text{.}\)
Solution.
Answers vary

Subsection 3.3.4 After Class Activities

Example 3.3.23.

Find the intervals on which \(\sin x + \cos x\) is increasing/decreasing on \([0,2\pi]\text{,}\) the local maximum and minimum values of the function on the interval, and the intervals of concavity/inflection points.
Solution.
The function is increasing on \(\lrpar{0,\dfrac{\pi}{4}}\cup \lrpar{\dfrac{5\pi}{4},2\pi}\text{,}\) decreasing on \(\lrpar{\dfrac{\pi}{4},\dfrac{5\pi}{4}}\text{,}\) concave up on \(\lrpar{\dfrac{3\pi}{4},\dfrac{7\pi}{4}}\text{,}\) and concave down on \(\lrpar{0,\dfrac{3\pi}{4}}\cup \lrpar{\dfrac{7\pi}{4},2\pi}\text{.}\) The local max occurs at \(\lrpar{\dfrac{\pi}{4},\sqrt{2}}\) and the local min occurs at \(\lrpar{\dfrac{5\pi}{4},-\sqrt{2}}\text{.}\)

Example 3.3.24.

Find the local maximum and minimum values of \(f(x) = \dfrac{x^2}{x-1}\) using the First and Second Derivative Tests.
Solution.
The local max is at \(x=0\) and the local min is at \(x=2\)

Example 3.3.25.

A graph is given below.
State and justify the \(x-\)coordinates of the inflection points of \(f\text{,}\) if
  1. The curve is the graph of \(f\)
  2. The curve is the graph of \(f'\text{.}\)
  3. The curve is the graph of \(f''\text{.}\)
Solution.
  1. \(\displaystyle x=3,5\)
  2. \(\displaystyle x=2,4,6\)
  3. \(\displaystyle x=1,7\)

Example 3.3.26.

Consider the phrase “SAT scores are declining at a slower rate” Interpret this statement in terms of a function and its first and second derivatives.
Solution.
\(f'(x) \gt 0, f''(x)\lt 0\)
<Prev^TopNext>