4.1 Extreme Values for a function Absolute Extreme Values (a)There is an absolute maximum value at x = c iff f(c)  f(x) for all x in the entire domain. (a)There is an absolute minimum value at x = c iff f(c)  f(x) for all x in the entire domain. Relative Extreme Values (a)There is a relative maximum value at x = c iff f(c)  f(x) for all x in some open interval containing c. (a)There is a relative minimum value at x = c iff f(c)  f(x) for all x in some open interval containing c.

Maxima and Minima

Where f (c) =0 Where f (c ) is undefined At an endpoint of a closed interval Locations of Extreme values **Values in the domain of f where f (c) is zero or is undefined are called critical values of the function.*** If f(x) has a maximum or minimum value at x = c it must occur at one of the following locations:

Maxima and Minima on closed interval for continuous function must exist.

Maximum and minimums

A curve with a local maximum value. The slope at c, is zero. Local Max

4.3 First Derivative test for Increasing and Decreasing functions If a function is continuous on [a, b] and differentiable on (a,b) (a) If f > 0 at each point of (a,b) then f increases on [a,b]. (a) If f < 0 at each point of (a,b) then f decreases on [a,b].

Find the critical points and identify intervals on which f is increasing and decreasing.

First derivative test for Local Extrema At a critical point x = c 1.f has a local minimum if f changes from negative to positive at c. 2.f has a local maximum if f changes from positive to negative at c. 3.There is no extreme value if the sign of f does not change. Could be a horizontal tangent without direction change.

Figure 3.24: The graph of f (x) = x 3 is concave down on (– , 0) and concave up on (0,  ). Concavity

Second derivative test for Concavity A graph is concave up on any interval where The second derivative is positive. A graph is concave down on any interval where The second derivative is negative. A point of inflection for a function Occurs where the concavity changes

Second derivative test for extreme values If there is a critical value at x = c and A local max at x = c conclusion A local min at x = c inconclusive

Section 1 / Figure 1 Section 4.3  Figures 11 Graph of the curve © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. A

The graph of f (x) = x 4 – 4x Finding intervals of concave up and concave down and Inflection points

What derivatives tell us about the shape of a graph

Figure 1.42: The blue graph of f (x) = x + e –x looks like the graph of g(x) = x (black) to the right of the y-axis and like the graph of h(x) = e –x (red) to the left of the y-axis. (Example 1) 4. 4 Limits to Infinity (End behavior) What happens to the value of the function when the value of x increases without bound? What happens to the value of the function when the value of x decreases without bound?

Basic limits to infinity

Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function gets closer and closer to 5/3.

Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function gets closer and closer to 0.

Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function decreases without bound.

Figure 1.27: The function in Example 3. As x gets larger and larger, the function gets closer and closer to 5/3. When the limit to infinity exists, at y = L we can say that the line y = L is a horizontal asymptote.

Figure 1.27: The function in Example 3. Horizontal Asymptote

Limits that are infinite (y increases without bound) An infinite limit will exist as x approaches a finite value when direct substituion produces If an infinite limit occurs at x = c we have a vertical asymptote with the equation x = c.

Figure 1.29: The function in Example 5(a). Slant Asymptote As x gets larger and larger, the graph gets closer and closer to the line As x gets smaller and smaller, the graph gets closer to the same line. You can use long division to rewrite the given function. There is a vertical asymptote at x = - 4 / 7

Section 1 / Figure 1 Graphs of the polynomial © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. 4.6Different scales

Section 1 / Figure 1 Derivatives of the polynomial © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning ™ is a trademark used herein under license. Graphs of derivatives

Section 1 / Figure 1

Section 4.6  Figures 19, 20 The family of functions