Precalculus Digital Learning Week of 5/18 Lesson 16 The Formal Definition of the Print these 3 pages if you can. The first 2 pages are discussed in the video, page 3 contains a couple problems for you to try. Video – Digital Learning Lesson 16 (a link to the videos can be found on my website)

Learning Targets: I can determine the derivative of functions using the formal definition. I can determine the equation of the line at points on polynomial functions.

The Difference Quotient of a

푓(푥 + ℎ) − 푓(푥) 퐷푄 = ℎ

represents the of the from any x to x+h for an incremental amount h.

As h gets smaller towards 0, our two points become 1 and we find the slope of the tangent line at x

Using limits ( ℎ → 0 ) we can determine the slope of a tangent line at most points on a graph. When we talk about the slope of a curve we will refer to the slope of the tangent line at a specific point. The Derivative is a formula to find the slope of a tangent line at any one point on the curve. This process leads to the Formal Definition of the Derivative.

Noatation: Given function 푦 = 푓(푥) the deivative can be denoted as 푓′(푥) or 푦′ or 푑푦 푑푥

The Formal Definition of a Derivative

( ) The derivative of function 푓(푥) is: 푓′(푥) = lim 푓 푥+ℎ −푓(푥) ℎ→0 ℎ

As a limit, direct substituion should be our first attemp to evaluate, here’s what will happen:

푓(푥 + ℎ) − 푓(푥) 푓(푥 + 0) − 푓(푥) 푓(푥) − 푓(푥) 0 푓′(푥) = lim = = = ℎ→0 ℎ 0 0 0

Fortunately, we know how to evaluate limits that yield a 0 result: cancel a common factor and retry ! 0 The Formal Definition of a Derivative

( ) The derivative of function 푓(푥) is: 푓′(푥) = lim 푓 푥+ℎ −푓(푥) ℎ→0 ℎ

Recall: Derivative is a formula to find the slope of a tangent line at anypoint one point on the curve.

1) a) Use the definition to find the derivative of 푓(푥) = 푥2 + 7푥

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) =

b) Determine the slope of that tangent line when 푥 = −1

2) a) Use the definition to find the derivative of 푔(푥) = 푥2 − 9푥 − 5

푔′(푥) = lim ℎ→0

푔′(푥) = lim ℎ→0

푔′(푥) = lim ℎ→0

푔′(푥) =

b) Determine the slope of that tangent line when 푥 = 2

c) Determine the equation of the tangent line when 푥 = 2

Pause the video for a few minutes and try questions 3, 4 &5 then resume to check your results. The Formal Definition of a Derivative

( ) The derivative of function 푓(푥) is: 푓′(푥) = lim 푓 푥+ℎ −푓(푥) ℎ→0 ℎ

3) a) Use the definition to find the derivative of 푓(푥) = 푥2 + 4

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) =

b) Determine the slope of that tangent line when 푥 = −1

4) a) Use the definition to find the derivative of 푓(푥) = 5푥 + 3

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) =

5) a) Use the definition to find the derivative of 푓(푥) = 푥2 + 3푥 + 4

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) = lim ℎ→0

푓′(푥) =