Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.9 – Antiderivatives
• Section 5.3 – The Fundamental Theorem of Calculus
• Section 5.4 – Indefinite Integrals and the Net Change Theorem

Prerequisite Knowledge

Calculus I

NOTE: One way to use the information contained within this document is to graphically verify antiderivatives.

Relative Extrema

Definition: A function $f(x)$$f(x)$ has a local (relative) maximum at $x=x_0$$x=x_0$ if and only if there is some interval $(a,b)$$(a,b)$ that contains $x_0$$x_0$ such that $f(x_0)\ge f(x)$$f(x_0)\geq f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$.

Definition: A function $f(x)$$f(x)$ has a local (relative) minimum at $x=x_0$$x=x_0$ if and only if there is some interval $(a,b)$$(a,b)$ that contains $x_0$$x_0$ such that $f(x_0)\le f(x)$$f(x_0)\leq f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$.

Definition: Critical numbers are location(s), i.e., $x$$x$-values, of potential local extrema.

• How to find locations, called critical numbers, of local (relative) extrema:

  • Analytically:

    • Solve $f^{\prime }(x)=0$$f^{\,\prime}(x)=0$ solve for $x$$x$-values; and
    • find $x$$x$-values such that $f^{\prime }(x)=\text{ dne}$$f^{\,\prime}(x)=\text{ dne}$.

  • Graphically:

    • Find $x$$x$-values where the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ touches or crosses the $x$$x$-axis; and

    • find $x$$x$-values where $f^{\prime }(x)=\text{ dne}$$f^{\,\prime}(x)=\text{ dne}$, i.e., where the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ has

      • a discontinuity; or
      • a cusp; or
      • a vertical tangent line.

  • NOTE: A critical number does not always refer to a local extremum.

First Derivative Test for Local Maxima

• Analytically:

  • $f(x)$$f(x)$ has a local maximum at $x=x_0$$x=x_0$ if

    • $f^{\prime }(x)>0$$f^{\,\prime}(x)>0$ for all $x$$x$ in some interval $(a,x_0)$$(a, x_0)$; and
    • $f^{\prime }(x)<0$$f^{\,\prime}(x)<0$ for all $x$$x$ in some interval $(x_0,b)$$(x_0, b)$.

• Graphically:

  • $f(x)$$f(x)$ has a local maximum at $x=x_0$$x=x_0$ if

    • the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ is above the $x$$x$-axis for all $x$$x$ in some interval $(a,x_0)$$(a, x_0)$; and
    • the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ is below the $x$$x$-axis for all $x$$x$ in some interval $(x_0,b)$$(x_0, b)$.

First Derivative Test for Local Minima

• Analytically:

  • $f(x)$$f(x)$ has a local maximum at $x=x_0$$x=x_0$ if

    • $f^{\prime }(x)>0$$f^{\,\prime}(x)>0$ for all $x$$x$ in some interval $(a,x_0)$$(a, x_0)$; and
    • $f^{\prime }(x)<0$$f^{\,\prime}(x)<0$ for all $x$$x$ in some interval $(x_0,b)$$(x_0, b)$.

• Graphically:

  • $f(x)$$f(x)$ has a local minimum at $x=x_0$$x=x_0$ if

    • the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ is below the $x$$x$-axis for all $x$$x$ in some interval $(a,x_0)$$(a, x_0)$; and
    • the graph of $f^{\prime }(x)$$f^{\,\prime}(x)$ is above the $x$$x$-axis for all $x$$x$ in some interval $(x_0,b)$$(x_0, b)$.

Intervals of Increase

• Analytically:

  • $f(x)$$f(x)$ is increasing at a point $x_0$$x_0$ if and only if there exists some interval $(a,b)$$(a,b)$ containing $x_0$$x_0$ such that $f(x_0)>f(x)$$f(x_0) > f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$ to the left of $x_0$$x_0$ and $f(x_0)<f(x)$$f(x_0) < f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$ to the right of $x_0$$x_0$; or
  • $f(x)$$f(x)$ is increasing at a point $x_0$$x_0$ if and only if $f^{\prime }(x_0)>0$$f^{\,\prime}(x_0) > 0$.

• Graphically:

  • $f(x)$$f(x)$ is increasing at a point $x_0$$x_0$ if and only if the graph of $f^{\prime }(x_0)$$f^{\,\prime}(x_0)$ is above the $x$$x$-axis.

Intervals of Decrease

• Analytically:

  • $f(x)$$f(x)$ is decreasing at a point $x_0$$x_0$ if and only if there exists some interval $(a,b)$$(a,b)$ containing $x_0$$x_0$ such that $f(x_0)<f(x)$$f(x_0) < f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$ to the left of $x_0$$x_0$ and $f(x_0)>f(x)$$f(x_0) > f(x)$ for all $x$$x$ in $(a,b)$$(a,b)$ to the right of $x_0$$x_0$; or
  • $f(x)$$f(x)$ is decreasing at a point $x_0$$x_0$ if and only if $f^{\prime }(x_0)<0$$f^{\,\prime}(x_0) < 0$.

• Graphically:

  • $f(x)$$f(x)$ is decreasing at a point $x_0$$x_0$ if and only if the graph of $f^{\prime }(x_0)$$f^{\,\prime}(x_0)$ is below the $x$$x$-axis.

Concavity

• A function $f(x)$$f(x)$ is concave up at $x_0$$x_0$ if and only if $f^{\prime }(x)$$f^{\,\prime}(x)$ is increasing at $x_0$$x_0$.

  • If $f^{\prime \prime }(x)$$f^{\,\prime\prime}(x)$ exists at $x_0$$x_0$ and is positive, then $f(x)$$f(x)$ is concave up at $x_0$$x_0$.

• A function $f(x)$$f(x)$ is concave down at $x_0$$x_0$ if and only if $f^{\prime }(x)$$f^{\,\prime}(x)$ is decreasing at $x_0$$x_0$.

  • If $f^{\prime \prime }(x)$$f^{\,\prime\prime}(x)$ exists at $x_0$$x_0$ and is negative, then $f(x)$$f(x)$ is concave down at $x_0$$x_0$.

Practice

Example 01: $h(x)=x^3-2x^2+3$$h(x)=x^3-2x^2+3$

• Find locations of all local extrema of $h(x)$$h(x)$, if exists.
• Find all local extrema of $h(x)$$h(x)$, if exists.
• Find interval(s) on which $h(x)$$h(x)$ is increasing, if exists.
• Find interval(s) on which $h(x)$$h(x)$ is decreasing, if exists.
• Find all inflection points of $h(x)$$h(x)$, if exists.
• Find interval(s) on which $h(x)$$h(x)$ is concave up, if exists.
• Find interval(s) on which $h(x)$$h(x)$ is concave down, if exists.

Example 02: $f(x)=x^2{e}^x$$f(x)=x^2 e^x$

• Find locations of all local extrema of $f(x)$$f(x)$, if exists.
• Find all local extrema of $f(x)$$f(x)$, if exists.
• Find interval(s) on which $f(x)$$f(x)$ is increasing, if exists.
• Find interval(s) on which $f(x)$$f(x)$ is decreasing, if exists.
• Find all inflection points of $f(x)$$f(x)$, if exists.
• Find interval(s) on which $f(x)$$f(x)$ is concave up, if exists.
• Find interval(s) on which $f(x)$$f(x)$ is concave down, if exists.

Example 03: $g(x)=x^2\sin (x)$$g(x)=x^2 \sin{(x)}$ on $[-2\pi ,\pi ]$$[-2\pi,\pi]$

• Find locations of all local extrema of $g(x)$$g(x)$, if exists.
• Find all local extrema of $g(x)$$g(x)$, if exists.
• Find locations of all absolute extrema of $g(x)$$g(x)$, if exists.
• Find all absolute extrema. How do you know $g(x)$$g(x)$ has absolute extrema?
• Find interval(s) on which $g(x)$$g(x)$ is increasing, if exists.
• Find interval(s) on which $g(x)$$g(x)$ is decreasing, if exists.
• Find all inflection points of $g(x)$$g(x)$, if exists.
• Find interval(s) on which $g(x)$$g(x)$ is concave up, if exists.
• Find interval(s) on which $g(x)$$g(x)$ is concave down, if exists.

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Created: Monday, 18 January 2021 09:33 EDT Last Modified: Monday, 15 August 2022 - 15:08 (EDT)