Presentation on theme: "4.3 exactly how Derivatives influence the shape of a Graph"— Presentation transcript:

1 4.3 just how Derivatives impact the form of a GraphAPPLICATIONS the DIFFERENTIATION. 4.3 how Derivatives affect the shape of a Graph In this section, we will learn: just how the derivative of a role gives us the direction in i m sorry the curve proceeds at every point.

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2 WHAT go f’ SAY about f ? to see how the derivative the f deserve to tell united state where a role is enhancing or decreasing, look at the figure. Raising functions and also decreasing features were characterized in section 1.1

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3 WHAT does f’ SAY about f ? in between A and B and between C and D, the tangent lines have actually positive slope. So, f "(x) > 0. Between B and C, The tangent lines Have negative slope. So, f "(x)

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0. In between B and C, The tangent lines. Have an adverse slope. So, f (x)

4 INCREASING/DECREASING check (I/D TEST)If f "(x) > 0 on an interval, climate f is boosting on that interval. If f "(x)

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0 on one interval, climate f is enhancing on the interval. If f (x)

5 I/D TEST instance 1 uncover where the role f(x) = 3x4 – 4x3 – 12x is increasing and also where that is decreasing.

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6 f "(x) = 12x3 - 12x2 - 24x = 12x(x – 2)(x + 1)I/D TEST instance 1 f "(x) = 12x3 - 12x2 - 24x = 12x(x – 2)(x + 1) To use the identifier Test, we have to know wherein f "(x) > 0 and also where f "(x) This relies on the indications of the three determinants of f "(x)—namely, 12x, x – 2, and also x + 1.

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0 and also where f (x)

7 I/D TEST instance 1 We divide the actual line into intervals who endpoints room the critical numbers -1, 0, and also 2 and also arrange our occupational in a chart. F "(x) = 12x(x – 2)(x + 1)

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8 WHAT go f " SAY about f ? recall from section 4.1 that, if f has a local maximum or minimum in ~ c, climate c must be a an essential number that f (by Fermat’s Theorem). However, not every crucial number gives rise come a maximum or a minimum. So, we need a check that will certainly tell united state whether or no f has actually a regional maximum or minimum at a vital number.

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9 WHAT walk f’ SAY around f ? You deserve to see from the figure that f(0) = 5 is a regional maximum worth of f because f increases on (-1, 0) and decreases on (0, 2). In regards to derivatives, f "(x) > 0 for -1 f "(x)

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0 for -1

10 Suppose that c is a an important number the a consistent function f.

You are watching: How derivatives affect the shape of a graph

FIRST DERIVATIVE TEST expect that c is a crucial number of a continuous function f. If f " alters from hopeful to an adverse at c, then f has actually a regional maximum at c.

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11 b. If f’ transforms from an adverse to positive at c, climate f has actually a localFIRST DERIVATIVE check b. If f’ changes from an unfavorable to positive at c, climate f has actually a local minimum at c.

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12 c. If f " walk not readjust sign in ~ c—for example, FIRST DERIVATIVE check c. If f " does not readjust sign in ~ c—for example, if f " is positive on both political parties of c or an adverse on both sides—then f has no neighborhood maximum or minimum at c.

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13 WHAT walk f’ SAY around f ? example 2 discover the local minimum and maximum values of the role f in example 1.

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14 WHAT go f’ SAY around f ? example 2 indigenous the graph in the solution to example 1, we check out that f "(x) transforms from an unfavorable to positive at -1. So, f(-1) = 0 is a neighborhood minimum worth by the an initial Derivative Test.

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15 Similarly, f " transforms from an adverse to confident at 2.WHAT walk f’ SAY around f ? example 2 Similarly, f " transforms from an adverse to optimistic at 2. So, f(2) = -27 is also a local minimum value.

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16 WHAT does f’ SAY around f ? example 2 As formerly noted, f(0) = 5 is a regional maximum value due to the fact that f "(x) transforms from optimistic to an adverse at 0.

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17 Find the local maximum and minimum values of the functionWHAT walk f’ SAY around f ? example 3 find the neighborhood maximum and also minimum values of the role g(x) = x + 2 sin x 0 ≤ x ≤ 2π

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18 To find the vital numbers the g, we differentiate:WHAT walk f’ SAY around f ? instance 3 To uncover the an essential numbers that g, us differentiate: g"(x) = cos x So, g’(x) = 0 when cos x = - ½. The options of this equation space 2π/3 and 4π/3.

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19 So, we analysis g in the following table.WHAT walk f’ SAY around f ? example 3 together g is differentiable everywhere, the only an essential numbers room 2π/3 and 4π/3. So, we analyze g in the complying with table.

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20 WHAT walk f’ SAY about f ? example 3 as g"(x) transforms from confident to an adverse at 2π/3, the very first Derivative check tells united state that there is a regional maximum in ~ 2π/3. The regional maximum worth is:

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21 Likewise, g"(x) changes from negative to positive at 4π/3.WHAT go f’ SAY around f ? example 3 Likewise, g"(x) changes from an adverse to optimistic at 4π/3. So, a local minimum worth is:

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22 The graph the g supports our conclusion.WHAT go f’ SAY about f ? example 3 The graph the g support our conclusion.

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23 The figure shows the graphs of 2 increasing functions on (a, b).WHAT go f "" SAY around f ? The figure shows the graphs of two increasing attributes on (a, b).

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24 WHAT does f’’ SAY about f ?Both graphs join allude A to suggest B, however they watch different due to the fact that they bending in various directions. How can we distinguish between these two species of behavior?

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25 WHAT does f’’ SAY around f ?Here, tangents to this curves have been attracted at number of points.

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26 CONCAVITY—DEFINITIONIf the graph that f lies above all of its tangents on one interval I, that is referred to as concave upward on I. If the graph of f lies below all of its tangents ~ above I, the is called concave downward on I.

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27 CONCAVITY The number shows the graph that a role that is concave upward (CU) on the intervals (b, c), (d, e), and also (e, p) and concave downward (CD) ~ above the intervals (a, b), (c, d), and (p, q).

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28 CONCAVITY native this figure, you deserve to see that, going from left to right, the slope of the tangent increases. This means that the derivative f " is an increasing duty and thus its derivative f "" is positive.

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29 So, f " decreases and also therefore f "" is negative.CONCAVITY Likewise, in this figure, the slope of the tangent decreases from left to right. So, f " decreases and also therefore f "" is negative. This reasoning have the right to be reversed and suggests the the following theorem is true.

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30 CONCAVITY test – 2nd DERIVATIVE TESTIf f ""(x) > 0 for every x in I, climate the graph that f is concave upward (MIN) top top I. If f ""(x)

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0 for all x in I, then the graph the f is concave upward (MIN) ~ above I. If f (x)

31 B, C, D, and P room points the inflection.

See more: Divinity Original Sin Enhanced Edition Co Op, Divinity: Original Sin

INFLECTION point B, C, D, and also P are points of inflection.

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32 INFLECTION allude In check out of the Concavity Test, there is a allude of inflammation at any point where the second derivative alters sign.

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33 SECOND DERIVATIVE TESTSuppose f "" is continuous near c. If f "(c) = 0 and also f ""(c) > 0, then f has actually a regional minimum at c. B. If f "(c) = 0 and f ""(c) maximum in ~ c.

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0, climate f has a local. Minimum in ~ c. B. If f (c) = 0 and also f (c)

34 WHAT does f’’ SAY around f ?Example 6 talk about the curve y = x4 – 4x3 v respect to concavity, point out of inflection, and local maxima and minima. Use this information to sketch the curve.

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35 WHAT does f’’ SAY about f ?Example 6 If f(x) = x4 – 4x3, then: f "(x) = 4x3 – 12x2 = 4x2(x – 3) f ""(x) = 12x2 – 24x = 12x(x – 2)

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36 WHAT go f’’ SAY around f ?Example 6 To uncover the critical numbers, we set f "(x) = 4x2(x – 3)= 0 and obtain x = 0 and also x = 3. To use the 2nd Derivative Test, us evaluate f "" in ~ these vital numbers: f ""(0) = f ""(3) = 36 > 0 Note: f ""(x) = 12x(x – 2)

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0. Note: f (x) = 12x(x – 2)">

37 WHAT does f’’ SAY around f ?Example 6 f ""(3) > 0 suggests f(3) = -27 is a neighborhood minimum. Since f ""(0) = 0, the second Derivative Test offers no information around the critical number 0.

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0 means f(3) = -27 is a local minimum. Due to the fact that f (0) = 0, the 2nd Derivative Test gives no information about the vital number 0.">

38 WHAT does f’’ SAY around f ?Example 6 However, since f "(x)

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39 WHAT does f’’ SAY around f ?Example 6 because that f ""(x) = 0 we have f ""(-1) > 0 and f ""(1) The suggest (0, 0) is an inflection point—since the curve alters from concave increase to concave bottom there. Note: f ""(x) = 12x(x – 2)

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0 and also f (1)

40 WHAT go f’’ SAY around f ?Example 6 for f ""(2) = 0 we have f ""(1) 0. Hence, (2, -16) is an inflammation point—since the curve changes from concave downward to concave increase there. Note: f ""(x) = 12x(x – 2)

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41 WHAT walk f’’ SAY around f ?Example 6 utilizing the regional minimum, the intervals that concavity, and also the inflammation points, we sketch the curve.

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42 The 2nd Derivative check is inconclusive once f ""(c) = 0.NOTE The 2nd Derivative test is inconclusive once f ""(c) = 0. In other words, at such a point, there can be a maximum, a minimum, or no (as in the example). The test also fails once f ""(c) does not exist. In such cases, the very first Derivative Test have to be used.

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