Wednesday, March 16, 2016MAT 145 Please review TEST #2 Results and see me with questions, corrections, and concerns.

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Presentation transcript:

Wednesday, March 16, 2016MAT 145 Please review TEST #2 Results and see me with questions, corrections, and concerns.

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145 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, March 16, 2016MAT 145 (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? (1)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, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145 Extreme Value Theorem: If: a function 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, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145 Fermat’s Theorem If: a function f has a local maximum or a local minimum at x = c, and if f ’(c) exists, then f ’(c) = 0.

Wednesday, March 16, 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.

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 2016MAT 145

Wednesday, March 16, 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 ….

Wednesday, March 16, 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.