Math 111 Lecture Notes

Transcription

1 A function f is even if for ever in the domain of f it holds that f( ) = f(). Visuall, an even function is smmetric about the -ais. A function f is odd if for ever in the domain of f it holds that f( ) = f(). Visuall, an odd function is smmetric about the origin. Eample 1. Two classic eamples of even and odd functions are f() = and g() = 3, respectivel, as shown in Figures 1 and below. Figure 1. Graph of = f() 6 Figure. Graph of = g() Algebraicall verif that f is an even function and that g is an odd function. 1

2 Eample. Algebraicall determine if the following functions are even, odd, or neither. (a) h() = 3 (c) f(t) = t (b) g(t) = 1 t 1 (d) f() = Figure 3. = h() Figure. = g(t) Figure 5. = f(t) Figure 6. = f() t t Instructor: A.E.Car Page of 8

3 Eample 3. Algebraicall determine if the function f defined b f() = 3 is even, odd, or neither. Group Work 1. Determine if the following functions are even, odd, or neither. (a) g() = + 5 (b) f() = Instructor: A.E.Car Page 3 of 8

4 A function f is increasing on an open interval I if for ever 1 and in I with 1 < we have f( ) > f( 1 ). A function f is decreasing on an open interval I if for ever 1 and in I with 1 < we have f( ) < f( 1 ). A function f is constant on an open interval I if for ever 1 and in I with 1 < we have f( ) = f( 1 ). Eample. Determine the following for the function f graphed in Figure 7. State each using interval notation. Figure 7. = f() (a) Increasing: (b) Decreasing: (c) Constant: (d) Domain of f: (e) Range of f: A function has a local maimum at c if there eists an open interval I containing c so that for all not equal to c in I, it holds that f() < f(c). The output f(c) is referred to as the local maimum of f. A function has a local minimum at c if there eists an open interval I containing c so that for all not equal to c in I, it holds that f() > f(c). The output f(c) is referred to as the local minimum of f. Eample 5. Use Figure 7 to answer the following: (a) Identif all local maimum values of f and state where the occur. (b) Identif all local minimum values of f and state where the occur. Instructor: A.E.Car Page of 8

5 Let f be a function defined on an interval I. A function has an absolute maimum at u if it holds that f() f(u) for all in the interval I. The output f(u) is referred to as the absolute maimum of f. A function has an absolute minimum at u if it holds that f() f(u) for all in the interval I. The output f(u) is referred to as the absolute minimum of f. Eample 6. Use Figure 8 to answer the following: Figure 8. = f() (a) Identif all absolute maimum values of f and state where the occur. (b) Identif all absolute minimum values of f and state where the occur. Group Work. Use Figure 9 to answer the following: (a) Identif all local maimum values of g and state where the occur. Figure 9. = g() (b) Identif all local minimum values of g and state where the occur. (c) Identif all absolute maimum values of g and state where the occur. (d) Identif all absolute minimum values of g and state where the occur. Instructor: A.E.Car Page 5 of 8

6 Concavit So far, we have looked at where a function is increasing and decreasing and where it attains maimum and minimum values. We will now stud the concept of concavit. This concept involves looking at the rate at which a function increases or decreases. The graph of a function f whose rate of change increases (becomes less negative or more positive as ou move left to right) over an interval is concave up on that interval. Visuall, the graph bends upward. The graph of a function f whose rate of change decreases (becomes less positive or more negative as ou move left to right) over an interval is concave down on that interval. Visuall, the graph bends downward. Figure 10. Concave UP Figure 11. Concave DOWN Eample 7. The function defined b f() = is concave up on its entire domain. Notice that it is decreasing on the interval (, 0) and increasing on the interval (0, ). The function defined b f() = is concave down on its entire domain. Notice that it is increasing on the interval (, 0) and decreasing on the interval (0, ). Figure 1. Graph of = Figure 13. Graph of = Instructor: A.E.Car Page 6 of 8

7 Eample 8. The graph of = h() is shown in Figure 1. Use this to answer the following. Figure 1. Graph of = h() (a) State the interval(s) where h is positive. (b) State the interval(s) where h is negative. (c) State the interval(s) where h is increasing. (d) State the interval(s) where h is decreasing. (e) State the interval(s) where h is concave up. (f) State the interval(s) where h is concave down. (g) State an absolute maimum or absolute minimum values for h and where the occur. (h) State an local maimum or local minimum values for h and where the occur. Instructor: A.E.Car Page 7 of 8

8 Eample 9. Graph the function defined b f() = using a graphing calculator. (a) Determine an appropriate window that shows the important features. (b) State an local maimums/minimums and where each occurs. (c) State the interval(s) where the function is increasing and where it is decreasing. (d) State the interval(s) where the function is concave up and where it is concave down. (e) State the -intercept(s) and -intercept. Group Work 3. Graph the function defined b k() = using a graphing calculator. (a) Determine an appropriate window that shows the important features. (b) State an local maimums/minimums and where each occurs. (c) State the interval(s) where the function is increasing and where it is decreasing. (d) State the interval(s) where the function is concave up and where it is concave down. (e) State the -intercept(s) and -intercept. Instructor: A.E.Car Page 8 of 8