Lesson 22

Original PDF

Page 1

Lesson 22

Today:

  • Rolles Theorem
  • Mean Value Theorem
  • First Derivative Test

Office Hours: MWF 130pm – 230pm, MATH 842

Announcements:

  • Exam 2 is Graded
    Average = \( 12.25 / 18 \) (\( 68\% \))
    Median = \( 13.5 / 18 \) (\( 75\% \))

Page 2

Warmup Example:

What are some common features of both the graphs

Graph of a continuous and differentiable function on Cartesian axes. The curve begins at the endpoint (-4, 3) and ends at (4, 3), both indicated with blue dots and joined by a horizontal dashed line at y = 3. The function rises to a local maximum at (-3, 6), labeled with dashed lines to x = -3 and y = 6 and marked with a red horizontal tangent line. It then decreases steeply through the x-axis to a local minimum at (1, -5), indicated with dashed lines to x = 1 and y = -5 and marked with a red horizontal tangent line, before rising back up to the endpoint (4, 3).
Visual Description: Graph of a continuous and differentiable function on Cartesian axes. The curve begins at the endpoint (-4, 3) and ends at (4, 3), both indicated with blue dots and joined by a horizontal dashed line at y = 3. The function rises to a local maximum at (-3, 6), labeled with dashed lines to x = -3 and y = 6 and marked with a red horizontal tangent line. It then decreases steeply through the x-axis to a local minimum at (1, -5), indicated with dashed lines to x = 1 and y = -5 and marked with a red horizontal tangent line, before rising back up to the endpoint (4, 3).
Graph of a symmetric, U-shaped parabolic curve on Cartesian axes. The curve extends from x = -2 to x = 2, with endpoints at (-2, 5) and (2, 5) marked by blue dots. A horizontal dashed line connects the endpoints at y = 5, and vertical dashed lines drop to -2 and 2 on the x-axis. The curve reaches its minimum at the vertex at x = 0, where a red horizontal tangent line is drawn along the x-axis.
Visual Description: Graph of a symmetric, U-shaped parabolic curve on Cartesian axes. The curve extends from x = -2 to x = 2, with endpoints at (-2, 5) and (2, 5) marked by blue dots. A horizontal dashed line connects the endpoints at y = 5, and vertical dashed lines drop to -2 and 2 on the x-axis. The curve reaches its minimum at the vertex at x = 0, where a red horizontal tangent line is drawn along the x-axis.
  1. Continuous
  2. Differentiable
  3. Same function value at end points

There is atleast one point inside with horizontal tangent
\(\hookrightarrow\) derivative \(= 0\)

Page 3

Cartesian coordinate graph of a piecewise linear function resembling an inverted absolute value function. The graph starts with a point on the negative x-axis, rises linearly to a sharp peak on the positive y-axis, and decreases symmetrically to a point on the positive x-axis with the same vertical value of zero.
Visual Description: Cartesian coordinate graph of a piecewise linear function resembling an inverted absolute value function. The graph starts with a point on the negative x-axis, rises linearly to a sharp peak on the positive y-axis, and decreases symmetrically to a point on the positive x-axis with the same vertical value of zero.

Continuous ✓

end points have same value ✓

Not differentiable at every point

Rolles Cannot be Applied

Cartesian coordinate graph of a discontinuous function defined on [a, b]. The curve starts at point a on the negative x-axis, increases concavely to an open circle on the positive y-axis, then drops discontinuously to a lower point on the y-axis, and descends linearly to point b on the positive x-axis where f(a) = f(b) = 0.
Visual Description: Cartesian coordinate graph of a discontinuous function defined on [a, b]. The curve starts at point a on the negative x-axis, increases concavely to an open circle on the positive y-axis, then drops discontinuously to a lower point on the y-axis, and descends linearly to point b on the positive x-axis where f(a) = f(b) = 0.

\( f(a) = f(b) \)

\( f \) is Not continuous

Not differentiable

Cartesian coordinate graph of a smooth curve on the interval [a, b]. The curve begins at point a on the negative x-axis where f(a) = 0, increases smoothly through the y-axis, and terminates at a point above b on the positive x-axis, connected down to b by a dashed vertical line, showing f(a) does not equal f(b).
Visual Description: Cartesian coordinate graph of a smooth curve on the interval [a, b]. The curve begins at point a on the negative x-axis where f(a) = 0, increases smoothly through the y-axis, and terminates at a point above b on the positive x-axis, connected down to b by a dashed vertical line, showing f(a) does not equal f(b).

continuous

+

Differentiable

\( f(a) \neq f(b) \)

Page 4

Rolles Theorem

Illustration of Rolle's Theorem on a Cartesian plane. A continuous blue curve representing the function f(x) is plotted on the interval from x = a to x = b. Dashed vertical lines drop to the x-axis at a and b, and a horizontal dashed blue line connects the endpoints showing that f(a) = f(b). Within the open interval (a, b), the curve rises to a local maximum, falls to a local minimum below the x-axis, and rises again to a higher local maximum before returning to f(b). At each of these three extrema, a horizontal red tangent line is drawn, demonstrating locations where the derivative f'(c) equals zero.
Visual Description: Illustration of Rolle's Theorem on a Cartesian plane. A continuous blue curve representing the function f(x) is plotted on the interval from x = a to x = b. Dashed vertical lines drop to the x-axis at a and b, and a horizontal dashed blue line connects the endpoints showing that f(a) = f(b). Within the open interval (a, b), the curve rises to a local maximum, falls to a local minimum below the x-axis, and rises again to a higher local maximum before returning to f(b). At each of these three extrema, a horizontal red tangent line is drawn, demonstrating locations where the derivative f'(c) equals zero.
  1. \(f(x)\) is continuous on \([a, b]\)
  2. \(f(x)\) is differentiable on \((a, b)\)
  3. \(f(a) = f(b)\)

then there exist at least one number \(c\), \(a < c < b\) such that

\[ f'(c) = 0 \]

Page 5

e.g.:
  1. Continuous on \( [0, 2] \) ✓
  2. Differentiable on \( (0, 2) \) ✓
  3. \( f(0) = 6 \)
    \( f(2) = 8 - 8 + 6 = 6 \)
    ✓ \( f(0) = f(2) \)

Can Apply Rolles theorem \( \rightarrow \) there is a number \( c \) b/w \( 0 \) & \( 2 \) such that

\[ f'(c) = 0 \] \[ f'(x) = 3x^2 - 4 = 0 \] \[ x = \frac{2}{\sqrt{3}} \quad \text{or} \quad x = \frac{-2}{\sqrt{3}} \]

\( c = \frac{2}{\sqrt{3}} \) satisfies Rolles theorem

Page 6

Example: \( f(x) = e^{x^2} \) on \( [-3, 3] \)

  1. Continuous
  2. Differentiable

\( f \) is a composition of exponential & polynomial.

  1. \( f(-3) = e^{(-3)^2} = e^9 \)
    \( f(3) = e^{3^2} = e^9 \)

\( f(3) = f(-3) \)

Can Apply Rolles theorem \( \bar{\text{w}} \) there \( c \) btw \(-3\) & \( 3 \) such that \( f'(c) = 0 \)

\[ f'(x) = e^{x^2} \cdot 2x = 0 \]

\( \rightarrow \quad c = 0 \) satisfies Rolles theorem.

Page 7

Mean Value Theorem

Geometric representation of the Mean Value Theorem on a 2D Cartesian coordinate system. A continuous, differentiable curve y = f(x) in blue is shown between endpoints (a, f(a)) and (b, f(b)). A purple secant line passes through both endpoints (a, f(a)) and (b, f(b)). A red tangent line touches the curve at an intermediate point (c, f(c)) and is drawn parallel to the purple secant line. Dashed projection lines drop vertically to points a, c, and b on the x-axis, and extend horizontally to f(a) and f(b) on the y-axis.
Visual Description: Geometric representation of the Mean Value Theorem on a 2D Cartesian coordinate system. A continuous, differentiable curve y = f(x) in blue is shown between endpoints (a, f(a)) and (b, f(b)). A purple secant line passes through both endpoints (a, f(a)) and (b, f(b)). A red tangent line touches the curve at an intermediate point (c, f(c)) and is drawn parallel to the purple secant line. Dashed projection lines drop vertically to points a, c, and b on the x-axis, and extend horizontally to f(a) and f(b) on the y-axis.

Slope of line through end points \( (a, f(a)) \) & \( (b, f(b)) \)

\[ = \frac{f(b) - f(a)}{b - a} \]

\( f(x) \) is continuous on \( [a, b] \)
\( f(x) \) is differentiable on \( (a, b) \)
then there is a number \( c \) btw \( a \) & \( b \) such that tangent line at \( (c, f(c)) \) is parallel to line through \( (a, f(a)) \), \( (b, f(b)) \)

\[ \leadsto f'(c) = \frac{f(b) - f(a)}{b - a} \]

Page 8

MVT

Suppose \( f(x) \) is continuous on \( [a, b] \)
differentiable on \( (a, b) \)
then there a number \( c \), \( a < c < b \)
such that

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Rolles is a special case of MVT with \( f(a) = f(b) \).

Page 9

Example: \( f(x) = \sin x \) on \( [0, \pi/2] \)

Cartesian coordinate graph illustrating the Mean Value Theorem for f(x) = sin(x) on the interval [0, pi/2]. A blue curve representing the sine function starts at the origin (0, 0) and reaches a maximum value of 1 at x = pi/2, marked with dashed blue guide lines to the axes. A purple secant line connects the endpoints (0, 0) and (pi/2, 1). A parallel red tangent line touches the blue sine curve at an intermediate point with horizontal coordinate c, indicated by a vertical dotted line from the curve to the x-axis between 0 and pi/2.
Visual Description: Cartesian coordinate graph illustrating the Mean Value Theorem for f(x) = sin(x) on the interval [0, pi/2]. A blue curve representing the sine function starts at the origin (0, 0) and reaches a maximum value of 1 at x = pi/2, marked with dashed blue guide lines to the axes. A purple secant line connects the endpoints (0, 0) and (pi/2, 1). A parallel red tangent line touches the blue sine curve at an intermediate point with horizontal coordinate c, indicated by a vertical dotted line from the curve to the x-axis between 0 and pi/2.

find \( c \) b/w \( 0 \ \& \ \pi/2 \)

such that

\( f'(c) = \) slope of line through end points

\[ = \frac{f(\pi/2) - f(0)}{\pi/2 - 0} = \frac{1}{(\pi/2)} = 2/\pi \]

\( f'(x) = \cos x \)

want \( c \) such that

\[ \cos(c) = \frac{2}{\pi} \] \[ 0 < c = \cos^{-1}\left(\frac{2}{\pi}\right) < \pi/2 \]

Page 10

Example: \( f'(x) \geqslant 5 \) on \( [1, 3] \)

\( f(1) = 7 \), given that \( f \) is continuous on \( [1, 3] \), differentiable on \( (1, 3) \), find minimum possible value of \( f(3) \)?

Apply MVT on \( [1, 3] \)

there is \( c \) btw 1, 3 such that

\[ f'(c) = \frac{f(3) - f(1)}{3 - 1} \]

given \( \geqslant 5 \)

\[ \frac{f(3) - 7}{2} \geqslant 5 \] \[ f(3) \geqslant 17 \]

- min. value of \( f(3) = 17 \) to be able to satisfy given conditions.

Page 11

What the Derivative tells us?

Cartesian coordinate plot of a function f(x) over the domain from x = 0 to x = 6. The curve begins at the origin, increases through a positive slope tangent to a local maximum at x = 2 with a horizontal tangent line representing f'(2) = 0. The curve then decreases through x = 3 with a negative slope tangent down to a local minimum at x = 4 with a horizontal tangent line representing f'(4) = 0. From x = 4, it increases through x = 5 with a positive slope tangent up to x = 6. Vertical dashed guidelines connect x = 2, x = 4, and x = 6 on the x-axis to their corresponding points on the curve.
Visual Description: Cartesian coordinate plot of a function f(x) over the domain from x = 0 to x = 6. The curve begins at the origin, increases through a positive slope tangent to a local maximum at x = 2 with a horizontal tangent line representing f'(2) = 0. The curve then decreases through x = 3 with a negative slope tangent down to a local minimum at x = 4 with a horizontal tangent line representing f'(4) = 0. From x = 4, it increases through x = 5 with a positive slope tangent up to x = 6. Vertical dashed guidelines connect x = 2, x = 4, and x = 6 on the x-axis to their corresponding points on the curve.
  • \( f'(2) = 0 \) and \( f'(4) = 0 \)
  • intervals where \( f \) is increasing? \( \uparrow \)

    \( (0, 2) \) and \( (4, 6) \)

    \( f' > 0 \)

  • intervals where \( f \) is decreasing? \( \downarrow \)

    \( (2, 4) \)

    \( f' < 0 \)

at \( x = 2 \rightsquigarrow \) Local peak \( \rightsquigarrow \) Local Max or Relative Max

\( x = 4 \rightsquigarrow \) Local/Relative min

Page 12

First Derivative Test

\( f(x) \) has a relative max at \( x = c \) if

Illustration of the first derivative test for a relative maximum. A concave-down curved blue path peaks at a critical point marked by a red dot. To the left, a red tangent line has a positive slope indicating an increasing function with f' > 0. To the right, a red tangent line has a negative slope indicating a decreasing function with f' < 0.
Visual Description: Illustration of the first derivative test for a relative maximum. A concave-down curved blue path peaks at a critical point marked by a red dot. To the left, a red tangent line has a positive slope indicating an increasing function with f' > 0. To the right, a red tangent line has a negative slope indicating a decreasing function with f' < 0.
  1. \( x = c \) is a critical point
  2. increasing on left (\( f' > 0 \))
  3. decreasing on right (\( f' < 0 \))

\( f(x) \) has a relative min at \( x = c \) if

Illustration of the first derivative test for a relative minimum. A concave-up curved blue path forms a local trough. On the left side, a red tangent line slopes downward indicating a decreasing function with f' < 0. On the right side, a red tangent line slopes upward indicating an increasing function with f' > 0.
Visual Description: Illustration of the first derivative test for a relative minimum. A concave-up curved blue path forms a local trough. On the left side, a red tangent line slopes downward indicating a decreasing function with f' < 0. On the right side, a red tangent line slopes upward indicating an increasing function with f' > 0.
  1. \( x = c \) is a critical point
  2. decreasing on left (\( f' < 0 \))
  3. increasing on right (\( f' > 0 \))

Page 13

Eg: \( f(x) = 3x^5 - 5x^3 \), find Relative max. and Relative min.

\[ f'(x) = 15x^4 - 15x^2 = 15x^2(x^2 - 1) = 15x^2(x - 1)(x + 1) \]

Critical points are \( x = 0 \), \( x = 1 \), \( x = -1 \)

A first derivative sign chart plotted along a horizontal number line for f'(x) = 15x^2(x - 1)(x + 1). Points on the number line are labeled at -1, 0, and 1. Above the line, the signs of the derivative are indicated: plus (+) for x < -1, minus (-) for -1 < x < 0, minus (-) for 0 < x < 1, and plus (+) for x > 1. Below the line, vertical arrows show function behavior: an upward arrow on x < -1, a downward arrow on -1 < x < 0, a downward arrow on 0 < x < 1, and an upward arrow on x > 1. The formula f' = 15x^2(x - 1)(x + 1) is annotated to the right of the axis.
Visual Description: A first derivative sign chart plotted along a horizontal number line for f'(x) = 15x^2(x - 1)(x + 1). Points on the number line are labeled at -1, 0, and 1. Above the line, the signs of the derivative are indicated: plus (+) for x < -1, minus (-) for -1 < x < 0, minus (-) for 0 < x < 1, and plus (+) for x > 1. Below the line, vertical arrows show function behavior: an upward arrow on x < -1, a downward arrow on -1 < x < 0, a downward arrow on 0 < x < 1, and an upward arrow on x > 1. The formula f' = 15x^2(x - 1)(x + 1) is annotated to the right of the axis.
  • \( x = -1 \) is local Max.
  • neither max nor min
  • \( x = 1 \) is local Min