Supplemental 1.5. Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient

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1 Supplemental 1.5 Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient Interval Notation Many times in this class we will only want to talk about what happens to a function over a certain interval of values. These values can be written using interval notation. This is essentially a shortcut to writing what you see on the number line in other words you see real numbers on a number line, so it tells you about all real numbers between two endpoints. It tells about the inclusion or exclusion of left endpoint ( Endpoints are the beginning or end of the solution set) and the right endpoint and slams them together with a comma in the middle. One method of describing all real numbers is the interval: (-, ). Infinity is elusive, since you can never reach it, and therefore in interval notation we always use a parenthesis around infinity. Summary Endpoint included [ or ] Endpoint not included ( or ) Negative Infinity (- Positive Infinity ) An interval with both endpoints included is denoted as a closed interval. An interval with both endpoints not included is called an open interval. Finally, an interval with one endpoint included and one not included is called a half-open interval. Closed interval [-5, 3] Open interval ( , 14) Half-Open Interval [0, 1050) (-0.2, 101] Note: It is sometimes difficult to tell the difference between an open interval and an ordered pair. To help relieve the confusion your book and I will try to use the point ( , 14) vs the interval ( , 14). Increasing & Decreasing Functions 1) Interval Increasing: For decreasing values of the independent the corresponding values of the dependent also decrease. As the x s go up so do the y s. When x 1 < x 2, then f(x 1 ) < f(x 2 ). Decreasing: For decreasing values of the independent the corresponding values of the dependent increase. As the x s go up the y s go down. When x 1 < x 2, then f(x 1 ) > f(x 2 ). 2) Graphical Increasing: Climbs up when viewed left to right

2 Decreasing: Slides down when viewed left to right Using interval notation, give the intervals for the independent variable on which the function is a) increasing. b) decreasing x What about a given function? How could you tell from an equation where a function increases or decreases? 1) You know a great deal about different types of functions and/or 2) You use technology to investigate the graph of a function Let s start by studying some distinct functions (all are power functions): Quadratic: y = ax 2 + bx + c Sign of a determines opens up or down "+" concave up & decreasing on x < vertex while increasing on x > vertex " " concave down & increasing on x < vertex while decreasing on x > vertex The vertex (where the graph changes from increasing to decreasing or vice versa) is: (- b / 2a, f(- b / 2a )) Symmetric around a vertical line called a line of symmetry Goes through the vertex: x = - b / 2a Vertex: (0, 0) y = ax 2 (0, 0) Concave Up Decrease x < 0 Increasing x > 0 Vertex: (0, 0) y = -ax 2 (0, 0) Concave Down Increasing x < 0 Decreasing x > 0

3 Cubic: y = ax 3 + bx 2 + cx + d Form a lazy "S" shaped graph Sign of "a" determines curves up and to right or down and to right "+" increasing & concave down on neg. x and concave down on pos. x " " decreasing & concave up on neg x and concave down on pos. x Point of Inflection is where the graph starts to have opposite slope (0, 0) in y = ax 3 (0, d) in y = ax 3 + d (c, d) in y = a(x c) 3 + d y = ax 3 Concavity Changes (0, 0) Increasing Concave Down x < 0 Concave Up x > 0 y = -ax 3 Concavity Changes (0, 0) Decreasing Concave Up x < 0 Concave Down x > 0 Square Root: y = a(x b) 1/2 + c Looks like half a parabola opening up or down from the x-axis Only half because domain is range of its inverse the quadratic, so the domain is (0, ) for the basic family graphed below Sign of "a" tells up or down from x-axis "+" is increasing & concave down " " is decreasing & concave up Vertex is (b, c) y = ax 1/2 Vertex (0, 0) Increasing Concave Down y = -ax 1/2 Vertex (0, 0) Decreasing Concave Up

4 Inverse: y = a(x b) -1 + c Look like a wide parabola in 1 st & 3 rd or in 2 nd & 4 th quadrants Domain must exclude zero Never touches or crosses an axis; either the x or y (forming asymptotes) Because x 0, but as it gets close it is very small which makes f(x) get big Like wise as x gets big, f(x) gets small but never gets to zero! Sign of "a" tells which quadrants "+" decreasing & concave down on x<0 & concave up on x>0 " " increasing & concave up on x<0 & concave down on x>0 y = ax -1 Decreasing & Concave Down x < 0 Decreasing & Concave Up x > 0 y = -ax -1 Increasing & Concave Up x < 0 Increasing & Concave Down x > 0 Now, let s practice finding increasing and decreasing regions: Identify where the regions where the function is increasing and decreasing using interval notation. Sketch a rough graph to justify your regions. a) f(x) = 4 x b) f(x) = x 2 4x c) f(x) = x + 1 d) f(x) = x 1 *e) f(x) = 1 / 4 x 4 2x 2 *Use your calculator to graph this function.

5 Average Rate of Change The rate of change is the slope of a linear function y = mx + b. There is zero rate of change for a constant function f(x) = b. You should be able to find the rate of change for any function given in any form (visual/tabular/formula/description). A slope tells us if the dependent is increasing or decreasing as the independent is increasing, thus a slope is telling us if a function is increasing or decreasing. positive slope negative slope increasing dependents decreasing dependents *It should be noted that the interval on which a function is increasing or decreasing may or may not include the point at which the function changes from increasing to decreasing. However, if we consider the largest interval on which the function is decreasing/increasing we will find ourselves including the point at which the change occurs. Unfortunately, not all functions are linear and the rate of change is not always constant. For non-linear functions we can find an average rate of change, which is the rate of change on an interval (in many applications this is an interval of time). This brings about the idea of a secant line a line created by two points on a curve that tells us about the average rate of change on a given interval. Ave. Rate of Change = f(b) f(a) = y b a x Visual of Average Rate of Change y B Secant Line A Secant Line a x = b a b y =f(b) f(a) t Slope if + then increasing function then decreasing function If in the picture above A is located at (1, 4) and B is located at (7, 16) find the average rate of change on the interval.

6 The height of the grapefruit t seconds after it is thrown in the air is shown by the table:*(p. 20 Table 1.9 Ex. 8 from Applied Calculus, Hughes- Hallet, et al, 4 th edition) t (sec) f(t) (ft) a) Calculate the average rate of change in the height of the grapefruit between 0 and 2 b) Calculate the average rate of change in the height of the grapefruit between 2 and 3 c) What do you notice about the average rates of change in a) & b)? Why are they different? What has caused this difference? d) Calculate the average rate of change in the height of the grapefruit between 4 and 5 e) Calculate the average rate of change in the height of the grapefruit between 5 and 6 f) What do you notice about the average rates of change in b) & d)? Why are they different? What has caused this difference? Compute the average rate of change for f(x) from x 1 to x 2. Round your answer to 2 decimal places where appropriate. Interpret your result graphically. f(x) = 1 / 2 x 2 5 on x = -1 to x = 4

7 Difference Quotient The difference quotient is said to represent the average rate of change of a function from point x to point x + h. The amount h in Calculus is shown to become smaller and smaller until it is nearly non-existent, giving us the concept of a tangent line from the familiar secant line. We are still creating a line between two points on a curve that tells us about the average rate of change on an interval Difference Quotient = f(x + h) f(x) = f(x + h) f(x) h 0 (x + h) x h The difference quotient gives us a formula to describe a constantly changing rate of change. This is very helpful for functions that do not have a constant rate of change. Let s investigate the difference formula for a linear and non-linear function. Use the difference formula to compute the slope of the function f(x) = 2x + 3 Does the computation agree with the slope of the line as you know it? Use the difference formula to compute the slope of the function g(x) = 1 / 2 x 2 5 We computed the average rate of change for g(x) between x = -1 and x = 4 on the last example on the previous page. Use your formula for the average rate of change for g(x) to show that it agrees with the last example on the previous page. You will use x = -1 and h = (4 (-1)) = 5 in the formula you just derived.