# 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\left(x\right)$$f(x)$ has a local (relative) maximum at $x={x}_{0}$$x=x_0$ if and only if there is some interval $\left(a,b\right)$$(a,b)$ that contains ${x}_{0}$$x_0$ such that $f\left({x}_{0}\right)\ge f\left(x\right)$$f(x_0)\geq f(x)$ for all $x$$x$ in $\left(a,b\right)$$(a,b)$.

Definition: A function $f\left(x\right)$$f(x)$ has a local (relative) minimum at $x={x}_{0}$$x=x_0$ if and only if there is some interval $\left(a,b\right)$$(a,b)$ that contains ${x}_{0}$$x_0$ such that $f\left({x}_{0}\right)\le f\left(x\right)$$f(x_0)\leq f(x)$ for all $x$$x$ in $\left(a,b\right)$$(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}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=0$$f^{\,\prime}(x)=0$ solve for $x$$x$-values; and
• find $x$$x$-values such that $f^{\,\prime}(x)=\text{ dne}$.
• Graphically:

• Find $x$$x$-values where the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ touches or crosses the $x$$x$-axis; and

• find $x$$x$-values where $f^{\,\prime}(x)=\text{ dne}$, i.e., where the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$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\left(x\right)$$f(x)$ has a local maximum at $x={x}_{0}$$x=x_0$ if

• ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)>0$$f^{\,\prime}(x)>0$ for all $x$$x$ in some interval $\left(a,{x}_{0}\right)$$(a, x_0)$; and
• ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)<0$$f^{\,\prime}(x)<0$ for all $x$$x$ in some interval $\left({x}_{0},b\right)$$(x_0, b)$.
• Graphically:

• $f\left(x\right)$$f(x)$ has a local maximum at $x={x}_{0}$$x=x_0$ if

• the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is above the $x$$x$-axis for all $x$$x$ in some interval $\left(a,{x}_{0}\right)$$(a, x_0)$; and
• the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is below the $x$$x$-axis for all $x$$x$ in some interval $\left({x}_{0},b\right)$$(x_0, b)$.

#### First Derivative Test for Local Minima

• Analytically:

• $f\left(x\right)$$f(x)$ has a local maximum at $x={x}_{0}$$x=x_0$ if

• ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)>0$$f^{\,\prime}(x)>0$ for all $x$$x$ in some interval $\left(a,{x}_{0}\right)$$(a, x_0)$; and
• ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)<0$$f^{\,\prime}(x)<0$ for all $x$$x$ in some interval $\left({x}_{0},b\right)$$(x_0, b)$.
• Graphically:

• $f\left(x\right)$$f(x)$ has a local minimum at $x={x}_{0}$$x=x_0$ if

• the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is below the $x$$x$-axis for all $x$$x$ in some interval $\left(a,{x}_{0}\right)$$(a, x_0)$; and
• the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is above the $x$$x$-axis for all $x$$x$ in some interval $\left({x}_{0},b\right)$$(x_0, b)$.

#### Intervals of Increase

• Analytically:

• $f\left(x\right)$$f(x)$ is increasing at a point ${x}_{0}$$x_0$ if and only if there exists some interval $\left(a,b\right)$$(a,b)$ containing ${x}_{0}$$x_0$ such that $f\left({x}_{0}\right)>f\left(x\right)$$f(x_0) > f(x)$ for all $x$$x$ in $\left(a,b\right)$$(a,b)$ to the left of ${x}_{0}$$x_0$ and $f\left({x}_{0}\right)$f(x_0) < f(x)$ for all $x$$x$ in $\left(a,b\right)$$(a,b)$ to the right of ${x}_{0}$$x_0$; or
• $f\left(x\right)$$f(x)$ is increasing at a point ${x}_{0}$$x_0$ if and only if ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left({x}_{0}\right)>0$$f^{\,\prime}(x_0) > 0$.
• Graphically:

• $f\left(x\right)$$f(x)$ is increasing at a point ${x}_{0}$$x_0$ if and only if the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left({x}_{0}\right)$$f^{\,\prime}(x_0)$ is above the $x$$x$-axis.

#### Intervals of Decrease

• Analytically:

• $f\left(x\right)$$f(x)$ is decreasing at a point ${x}_{0}$$x_0$ if and only if there exists some interval $\left(a,b\right)$$(a,b)$ containing ${x}_{0}$$x_0$ such that $f\left({x}_{0}\right)$f(x_0) < f(x)$ for all $x$$x$ in $\left(a,b\right)$$(a,b)$ to the left of ${x}_{0}$$x_0$ and $f\left({x}_{0}\right)>f\left(x\right)$$f(x_0) > f(x)$ for all $x$$x$ in $\left(a,b\right)$$(a,b)$ to the right of ${x}_{0}$$x_0$; or
• $f\left(x\right)$$f(x)$ is decreasing at a point ${x}_{0}$$x_0$ if and only if ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left({x}_{0}\right)<0$$f^{\,\prime}(x_0) < 0$.
• Graphically:

• $f\left(x\right)$$f(x)$ is decreasing at a point ${x}_{0}$$x_0$ if and only if the graph of ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left({x}_{0}\right)$$f^{\,\prime}(x_0)$ is below the $x$$x$-axis.

#### Concavity

• A function $f\left(x\right)$$f(x)$ is concave up at ${x}_{0}$$x_0$ if and only if ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is increasing at ${x}_{0}$$x_0$.

• If ${f}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(x\right)$$f^{\,\prime\prime}(x)$ exists at ${x}_{0}$$x_0$ and is positive, then $f\left(x\right)$$f(x)$ is concave up at ${x}_{0}$$x_0$.
• A function $f\left(x\right)$$f(x)$ is concave down at ${x}_{0}$$x_0$ if and only if ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$f^{\,\prime}(x)$ is decreasing at ${x}_{0}$$x_0$.

• If ${f}^{\phantom{\rule{0.167em}{0ex}}\prime \prime }\left(x\right)$$f^{\,\prime\prime}(x)$ exists at ${x}_{0}$$x_0$ and is negative, then $f\left(x\right)$$f(x)$ is concave down at ${x}_{0}$$x_0$.

#### Practice

Example 01: $h\left(x\right)={x}^{3}-2{x}^{2}+3$$h(x)=x^3-2x^2+3$

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

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

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

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

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