Calculus I (MAT 145) Dr. Day Wednesday Oct 25, 2017 Using Derivatives: Function Characteristics & Applications (Ch 4) Extreme Values (4.1) Wednesday, October 25, 2017 MAT 145

Related-Rates Strategy Identify variables, given values and rates, and requested rate at a specific moment in time (perhaps indicated by specific value of variable). Relate given variables. Implicitly differentiate to find relationship among the rates. If necessary, solve associated problem to find needed auxiliary relationships. Substitute and solve for requested rate at specific time. Report and interpret requested rate as increasing/decreasing and provide units! Wednesday, October 25, 2017 MAT 145

Related Rates: Assessing Your Work Define all variables, by explaining what each variable you are using represents. Include a sketch, labeled with variables or algebraic expressions that include variables. Identify all rate(s) included in this problem, and, if known, provide their values, and their units. Show full evidence of your calculus-based reasoning and steps to solve the problem, including calculations or calculator input and output. Write a sentence to interpret the results of your calculations, including units.   __ 1 pt: related sketch, drawing, graph; __ 1 pt: related-variables equation; __ 1 pt: related-rates equation; __ 1 pt: use known info; __ 3 pts: correct & correctly labeled result, with units, written as a sentence Wednesday, October 25, 2017 MAT 145 written as a sentence labeled result, with units, ____ 3 pts: correct and correctly ____ 1 pt: use known info; ____ 1 pt: related-rates equation; ____ 1 pt: related-variables equation; ____ 1 pt: related sketch, drawing, graph;   /7 TOTAL

Related Rates (3.9) Illustrations and Examples Temple University RR site (discussion and animation) Kelly’s RR notes (solutions and animations) Melting Snowball discussion and animation) More Temple University animations Paul’s Online Math Notes: RR (worked examples) More Relatives RR (worked examples) Wednesday, October 25, 2017 MAT 145

Ladder problem A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Wednesday, October 25, 2017 MAT 145

Conical water tank A water tank has the shape of an inverted cone with a base radius 2 m and height 4 m. If water is being pumped into the tank at 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep. Wednesday, October 25, 2017 MAT 145

Two cars heading toward intersection Car A is traveling west at 50 mi/hr and car B is traveling north at 60 mi/hr. Both are headed for the intersection of the two roads. At what rate are the cars heading for each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? Note: Read “rate cars heading for each other” as “rate at which direct line distance between them is decreasing.” Wednesday, October 25, 2017 MAT 145

Extreme Values (4.1) Sketch a continuous function y = f(x) on the closed interval −3 ≤ x ≤ 7, starting at A=(−3,4) and stopping at B=(7,−4). Do not sketch a straight segment! Wednesday, October 25, 2017 MAT 145

Extreme Values (4.1) At what x-value locations, if any, on the closed interval −3 ≤ x ≤ 7, does your function y = f(x) reach a maximum? A minimum? At those locations you just identified, what is the value of f ’ (x)? Are there any situations for which that would not be true? Wednesday, October 25, 2017 MAT 145

Absolute (global) Extremes and Relative (local) Extremes Wednesday, October 25, 2017 MAT 145

Absolute (Global) Extrema In the graph, the Absolute (Global) Maximum is 5 (y-value) and is found at x = 3. The value 5 is the greatest value of the function over its entire domain [1,7]. The Absolute (Global) Minimum is 2 and is found at x = 6. The value 2 is the least value of the Wednesday, October 25, 2017 MAT 145

Absolute Extrema What is the absolute maximum in the graph? (This means “Tell me the greatest y-value.”) Where is the absolute maximum located? (This means, “Tell me the x-value that corresponds to the maximum y-value.”) What is the absolute minimum in the graph? Where is the absolute minimum located? What do you notice about the locations of absolute extrema? Where could they occur? Wednesday, October 25, 2017 MAT 145

Extreme Value Theorem: If f is continuous on the closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]. Wednesday, October 25, 2017 MAT 145

Relative (Local) Extrema In the graph, Relative (Local) Maxima are: f(b) (y-value) found at x=b f(d) (y-value) found at x=d These values are greater than all the other y-values in a small neighborhood immediately to the right and left. The Relative (Local) Minima are: f(c) (y-value) found at x=c f(e) (y-value) found at x=e These values are less than all the other y-values nearby. Wednesday, October 25, 2017 MAT 145

Relative (Local) Extrema Where could relative extrema occur? Critical numbers are the locations where local extrema could occur. Critical points are the points (x- and y-values) that describe both the locations and function values at those points. Determine critical numbers for 1. 2. 3. Wednesday, October 25, 2017 MAT 145

Fermat’s Theorem Wednesday, October 25, 2017 MAT 145

Absolute Extrema—Closed Interval Method Wednesday, October 25, 2017 MAT 145

Absolute Extrema Where and what are the absolute and local extrema? Wednesday, October 25, 2017 MAT 145

Absolute Extrema Where and what are the absolute and local extrema? Wednesday, October 25, 2017 MAT 145

Absolute and Relative Extremes Absolute (Global) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) of all possible outputs. Relative (Local) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) in some small neighborhood along the x-axis. Extreme Value Theorem: For f(x) continuous on a closed interval, there must be extreme values. If f is continuous on a closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]. Fermat’s Theorem: If f has a local max or min at x = c and if f ’(c) exists then f ’(c) = 0. Critical Point: An interior point (not an endpoint) on f(x) with f ’(x) = 0 or f ’(x) undefined. Note: The function MUST EXIST at x = c for a critical point to exist at x = c. Wednesday, October 25, 2017 MAT 145

Absolute and Relative Extremes Ways to Find Extrema Local Extremes: examine behavior at critical points Absolute Extremes: examine behavior at critical points and at endpoints Example Determine critical numbers, absolute extrema, and relative extrema for the unrestricted function (all possible domain values) and then for the restricted domain [−1,3]. Wednesday, October 25, 2017 MAT 145

Wednesday, October 25, 2017 MAT 145

Wednesday, October 25, 2017 MAT 145

Wednesday, October 25, 2017 MAT 145