Monday, March 28, 2016MAT 145. Monday, March 28, 2016MAT 145 For f(x) and g(x) shown below, use all you calculus and pre-calculus knowledge to calculate.

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Presentation transcript:

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145 For f(x) and g(x) shown below, use all you calculus and pre-calculus knowledge to calculate and identify important information about each function and then graph each function. Do not use a calculator to graph!

Monday, March 28, 2016MAT 145 To determine the absolute maximum and absolute minimum values of a continuous function f on a closed interval [a,b], carry out these steps. (1)Determine all critical numbers of the function f on a < x < b. (2)Determine the value of the function f at each critical number. (3)Determine the value of f at each endpoint of the closed interval [a,b]. (4)Now compare outputs: The largest of the values calculated in steps (2) and (3) is the absolute maximum value; the smallest of these values in the absolute minimum.

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145 m a t h

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145 Concavity Animations More Concavity Animations

Monday, March 28, 2016MAT 145 Concavity Animations More Concavity Animations

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT An object is moving in a positive direction when …. 2.An object is moving in a negative direction when …. 3.An object speeds up when …. 4.An object slows down when …. 5.An object changes directions when …. 6.The average velocity over a time interval is found by …. 7.The instantaneous velocity at a specific point in time is found by …. 8.The net change in position over a time interval is found by …. 9.The total distance traveled over a time interval is found by ….

Monday, March 28, 2016MAT An object is moving in a positive direction when v(t) > 0. 2.An object is moving in a negative direction when v(t) < 0. 3.An object speeds up when v(t) and a(t) share same sign. 4.An object slows down when v(t) and a(t) have opposite signs. 5.An object changes directions when v(t) = 0 and v(t) changes sign. 6.The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!). 7.The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time. 8.The net change in position over a time interval is found by calculating the difference in the positions at the start and end of the interval. 9.The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements.

Monday, March 28, 2016MAT 145 Indeterminate forms: 0/0, ∞/∞, 0∞, ∞–∞, 1 ∞, 0 0, ∞ 0 l’Hospital’s Rule provides us with a mechanism for evaluating limits of indeterminate forms. If f(x)/g(x) is an indeterminate form of type 0/0 or ∞/∞, then: If a limit is in another one of the indeterminate forms, use algebra to manipulate expression into 0/0 or ∞/∞, then use L’Hospital’s Rule.

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145 Remember: Confirm that limit meets criteria for L’Hospital’s Rule If not, use algebra, until limit does meet criteria Use L’Hospital’s Rule If needed, use L’Hospital’s rule more than once

Monday, March 28, 2016MAT 145 LIMIT LAWS! Suppose that c is a constant and the limits and exist.

Monday, March 28, 2016MAT 145 MORE LIMIT LAWS! New!

Monday, March 28, 2016MAT 145

Monday, March 28, 2016MAT 145 How do we evaluate these limits? l’Hospital’s Rule provides us with a mechanism for evaluating limits of indeterminate forms. If f(x)/g(x) is an indeterminate form of type 0/0 or ∞/∞, then: