Warm-Up! 𝑓(𝑥) is a continuous function defined on the interval [−2,2]. 𝑓 −2 =−3, and 𝑓 2 =7. Which of the following must be true. There exists a value 𝑥=𝑎, where 𝑓 𝑎 =0 𝑓′(𝑥)>0 for all values of 𝑥 There exists a value 𝑥=𝑏, where 𝑓 𝑏 =−5 I. III. I. II. II. III. I. Not enough information to determine.

Derivative FRQ Practice

Given the table answer the questions Use the data in the table to approximate 𝑓 ′ −4.5 and 𝑓′(−2). Is there a value 𝑥, −3≤𝑥≤−4 at which 𝑓 ′ 𝑥 =−5? Justify! The function 𝑓 𝑥 =−.64 𝑥 3 −4.9 𝑥 2 −15.37𝑥−6 models the table of values. Find the rate of change when 𝑥=−1 𝒙 -2 -3 -4 -5 -6 -7 𝑓(𝑥) 10.2 13.5 18.5 27.9 49.3 81.7

The functions 𝑓 is defined by 𝑓 𝑥 = 5− 𝑥 2 2 Find 𝑓 ′ 𝑥 Write an equation for the line tangent to the graph of 𝑓 at 𝑥=−1 let 𝑔 be the function defined by 𝑔 𝑥 = 𝑓(𝑥), 1≤𝑥<3 &𝑥+10, 3≤𝑥≤7 , Is 𝑔 continuous at 3? Why?

𝒙 𝟒 𝟒<𝒙<𝟓 𝟓 𝟓<𝒙<𝟔 𝟔 𝟔<𝒙<𝟕 𝟕 𝑓(𝑥) 7 Positive 2 4 𝑓′(𝑥) −3 Negative 𝑔(𝑥) 9 −4 𝑔′(𝑥) −7 −2 The twice differentiable functions 𝑓 and 𝑔 are defined for all real numbers 𝑥. Values of 𝑓, 𝑓 ′ , 𝑔, and 𝑔′ for various values of 𝑥 are given in the table above. Find the 𝑥−coordinate of each relative minimum of 𝑓 on the interval [4,7]. Justify your answers. Explain why there must be a value 𝑐, for 5<𝑐<6, such that 𝑓 ′′ 𝑐 =0 Function ℎ is defined by ℎ 𝑥 = 𝑓 𝑥 +3 2 . Find ℎ′(2). Show the computations that lead to your answer.

Differentiate the function 𝑓 𝑥 = sin 3𝑥 + cos 𝑥 −3 sin 2 𝑥 2