First and Second Derivatives Tests AP Calculus AB

Warm-up Slides

Warm-up Monday, March 2 – C-Day Let f be the function given by 𝑓 𝑥 = 2𝑥−1 5 𝑥+1 . Which of the following is an equation for the line tangent to the graph of f at the point where x = 1? y = 21x + 2 y = 21x – 19 y = 11x – 9 y = 10x + 2 y = 10x – 8 #6 Practice Exam – Non-Calculator 𝑓 ′ 𝑥 = 2𝑥−1 5 1 + 𝑥+1 5 2𝑥−1 4 2 𝑓 ′ 𝑥 = 2𝑥−1 5 +10 𝑥+1 2𝑥−1 4 𝑓 ′ 1 = 2∙1−1 5 +10 1+1 2∙1−1 4 =1+20=21=𝑚 Therefore it is either A or B. 𝑓 1 = 2∙1−1 5 1+1 =2 The point is (1, 2) not (0, 2) therefore it cannot be A. 𝑦−2=21 𝑥−1 𝑦=21𝑥−21+2=21𝑥−19 B

Warm-up Tuesday, March 3 – A-Day The function f given by 𝑓 𝑥 =2 𝑥 3 −3 𝑥 2 −12𝑥 has a relative minimum at x = −1 2 3− 105 4 3+ 105 4 #5 Practice Exam – Non Calculator Find the derivative and set equal to 0 to find the critical points 𝑓 ′ 𝑥 =6 𝑥 2 −6𝑥−12=0 6 𝑥+1 𝑥−2 =0 𝑥=−1,2 Set up intervals and test points Interval (-∞, -1) (-1, 2) (2, ∞) Test point -2 0 3 f‘(x) (-)(-) = + (+)(-) = - (+)(+) = + Inc/Dec Increasing Decreasing Increasing Max Min C.

We have taken for granted that a function 𝑓 𝑥 is increasing if 𝑓 ′ 𝑥 is positive and decreasing if 𝑓 ′ 𝑥 is negative. We will now develop a method for finding and testing critical points to find relative/local extrema and intervals in which the graph is increasing and decreasing.

Theorem – The Sign of the Derivative Let f be a differentiable function on an open interval 𝑎, 𝑏 . If 𝑓 ′ 𝑥 >0 for 𝑥∈ 𝑎, 𝑏 , then f is increasing on 𝑎, 𝑏 If 𝑓 ′ 𝑥 <0 for 𝑥∈ 𝑎, 𝑏 , then f is decreasing on 𝑎, 𝑏

There is a useful test for determining whether a critical point is a min or max (or neither) based on the sign change of the derivative.

Theorem – First Derivative Test for Critical Points Assume that 𝑓 𝑥 is differentiable and let c be a critical point of 𝑓 𝑥 . Then 𝑓 ′ 𝑥 changes from + to – at c ⇒ 𝑓 𝑐 is a local maximum 𝑓 ′ 𝑥 changes from – to + at c ⇒ 𝑓 𝑐 is a local minimum To carry out the First Derivative Test, we make a useful observation: 𝑓 ′ 𝑥 can change sign at a critical point, but it cannot change sign on the interval between two consecutive critical points.

Example Determine the intervals in which the function is increasing and decreasing and any local extrema: 𝑓 𝑥 = 𝑥 3 −27𝑥−20

Increasing/Decreasing Example Determine the intervals in which the function is increasing and decreasing and any local extrema: 𝑓 𝑥 = 𝑥 3 −27𝑥−20 Find the derivative 𝑓 ′ 𝑥 =3 𝑥 2 −27 Find the critical points by setting the derivative equal to 0 3 𝑥 2 −27=0 ⇒ 𝑥=±3 Set up intervals using the critical points and test a value within the interval Interval −∞,−𝟑 −𝟑,𝟑 𝟑,∞ Test point -4 4 𝑓 ′ test point 3 −4 2 −27=21 3 0 2 −27=−27 3 4 2 −27=21 Increasing/Decreasing Increasing Decreasing Extrema Max at 𝑥=−3 Min at 𝑥=3

Another important property is concavity, which refers to the way the graph bends. When 𝑓 𝑥 is concave up, 𝑓 ′ 𝑥 is increasing – the slopes of the tangent lines increase. When 𝑓 𝑥 is concave down, 𝑓 ′ 𝑥 is decreasing – the slopes of the tangent lines decrease.

Definition – Concavity Let 𝑓 𝑥 be a differentiable function on an open interval 𝑎, 𝑏 . Then f is concave up on 𝑎, 𝑏 if 𝑓 ′ 𝑥 is increasing on 𝑎, 𝑏 and thus 𝑓 ′′ 𝑥 >0 f is concave down on 𝑎, 𝑏 if 𝑓 ′ 𝑥 is decreasing on 𝑎, 𝑏 and thus 𝑓 ′′ 𝑥 <0

Theorem – Test for Concavity Assume that 𝑓 ′′ 𝑥 exists for all 𝑥∈ 𝑎, 𝑏 If 𝑓 ′′ 𝑥 >0 for all 𝑥∈ 𝑎, 𝑏 , then f is concave up on 𝑎, 𝑏 If 𝑓 ′′ 𝑥 <0 for all 𝑥∈ 𝑎, 𝑏 , then f is concave down on 𝑎, 𝑏 Theorem – Inflection Points Assume that 𝑓 ′′ 𝑥 exists. If 𝑓 ′′ 𝑐 =0 and 𝑓 ′′ 𝑥 changes sign at x = c, then 𝑓 𝑥 has a point of inflection at x = c.

Example Find the points of inflection and intervals of concavity for the function: 𝑓 𝑥 =3 𝑥 5 −5 𝑥 4 +1

Find the second derivative Example Find the points of inflection and intervals of concavity for the function: 𝑓 𝑥 =3 𝑥 5 −5 𝑥 4 +1 Find the second derivative 𝑓 ′ 𝑥 =15 𝑥 4 −20 𝑥 3 𝑓 ′′ 𝑥 =60 𝑥 3 −60 𝑥 2 Find the possible points of inflection by setting the second derivative equal to 0. 60 𝑥 3 −60 𝑥 2 =0 ⇒ 𝑥=0, 1 Set up intervals using the possible points of inflection and test a value within the interval Interval −∞,𝟎 𝟎,𝟏 𝟏,∞ Test point -1 0.5 2 𝑓 ′′ test point 60 −1 3 −60 −1 2 =−120 60 0.5 3 −60 0.5 2 =−7.5 60 2 3 −60 2 2 =240 Concavity Concavity Down Concave Down Concave Up POI No POI at x = 0 POI at x = 1

Although nowadays almost all graphs are produced by computer, sketching graphs by hand is a useful way of solidifying your understanding of the basic concepts of the first and second derivative tests. Most graphs are made up of smaller arcs that have one of the four basic shapes, corresponding to the four possible sign combinations of 𝑓 ′ and 𝑓 ′′ . In graph sketching, we focus on the transition points, where the basic shape changes due to a sign change in either 𝑓 ′ or 𝑓 ′′ .

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = 𝑥 2 −4𝑥+3

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = 1 3 𝑥 3 − 1 2 𝑥 2 −2𝑥+3

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 =3 𝑥 4 −8 𝑥 3 +6 𝑥 2 +1

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = cos 𝑥 + 1 2 𝑥 over 0, 𝜋

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 =𝑥 𝑒 𝑥

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = 3𝑥+2 2𝑥−4

Example Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = 1 𝑥 2 −1

QUIZ Find the intervals in which the function is increasing or decreasing, the intervals in which the function is concave up or concave down, the local extrema, and the points of inflection. Use these to sketch the graph of the function. 𝑓 𝑥 = 3 𝑥 2 𝑥 2 −1