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Unit 2: Derivatives

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Today’s Learning Goals …

2.1 The Derivative Function Today’s Learning Goals … After getting you all excited, today we will be doing EXACTLY the same thing that we did all of last unit except now we will call that big long formula, THE DERIVATIVE, and define it as a function, f’(x). Oh, and we’ll determine when a function is differentiable … woohoo! BUT … I PROMISE THIS IS THE LAST DAY!!!! TOMORROW THE FUN BEGINS!!!

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The Derivative Function

The derivative of 𝑓(𝑥) with respect to 𝑥 is the function 𝑓 ′ 𝑥 , where 𝑓 ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ , provided that the limit exists. So, we are basically using the same formula as Unit 1 but we are finding the derivative for an arbitrary value of 𝑥. This is called the first principles.

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Example #1: Determine the derivative 𝑓 ′ 𝑥 of the function 𝑓 𝑥 = 𝑥 2 at an arbitrary value of 𝑥 using first principles. Determine the slopes of the tangents to the parabola 𝑦= 𝑥 2 at 𝑥=−2, 0, and 1.

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Example #2: Determine the equation of the normal to the graph of 𝑓 𝑥 = 1 𝑥 at 𝑥=2. The normal the graph of 𝑓 at a point 𝑃 is the line that is perpendicular to the tangent at 𝑃.

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The Existence of Derivatives

A function 𝑓 is said to be differentiable at 𝑎 if 𝑓 ′ 𝑎 exists At the points where 𝑓 is not differentiable, we say that the derivative does not exist. Three common ways for a derivative to fail to exit are shown. Cusp Vertical Tangent Discontinuity

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Example #3: Show that the absolute value function 𝑓 𝑥 = 𝑥 is not differentiable at 𝑥=0. Solution: lim ℎ→ 0 − ℎ ℎ = lim ℎ→ 0 − −ℎ ℎ = lim ℎ→ 0 − (−1) =−1 𝑓 ′ 0 = lim ℎ→0 𝑓 0+ℎ −𝑓(0) ℎ lim ℎ→ ℎ ℎ = lim ℎ→ ℎ ℎ = lim ℎ→ (1) =1 = lim ℎ→ ℎ −0 ℎ lim ℎ→ 0 − ℎ ℎ ≠ lim ℎ→ ℎ ℎ = lim ℎ→0 ℎ ℎ You can see that the graph has a “corner” at (0,0), which prevents a unique tangent from being drawn there. 𝑓 ′ 0 = lim ℎ→0 ℎ ℎ 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 ∴ Recall, ℎ = ℎ ,ℎ>0 −ℎ , ℎ<0 the derivative does not exist at 𝑥=0.

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What can we take away from this example …

Is it possible for a function to be continuous at a point and yet not differentiable? YES If a function is differentiable at a point, is it also continuous at this point? YES SO … CONTINUOUS ⇏ DIFFERENTIABLE BUT … DIFFERENTIABLE ⇒ CONTINUOUS

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Other Notation for Derivatives

𝑦′ for functions defined using 𝑦, i.e. 𝑦= 𝑥 𝑦 ′ =2𝑥 𝑑𝑦 𝑑𝑥 read “dee 𝑦 by dee 𝑥”, i.e. 𝑦= 𝑥 𝑑𝑦 𝑑𝑥 =2𝑥 Leibniz notation or we can omit 𝑦 and 𝑓 altogether i.e. 𝑑 𝑑𝑥 ( 𝑥 2 )=2𝑥 𝑑𝑦 𝑑𝑥 = lim ∆𝑥→0 ∆𝑦 ∆𝑥

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In summary … QUESTIONS: p.73-75 #1, 2, 5cd, 6cd, 7bc, 11, 12

We defined the derivative function as, 𝑓 ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ , provided the limit exists. We learned that there can be points on a graph where the derivative does not exist and that: DIFFERENTIABILITY ⇒ CONTINUITY CONTINUITY ⇏ DIFFERENTIABLITY BUT Notations for the derivative include 𝑓 ′ 𝑥 , 𝑦 ′ 𝑎𝑛𝑑 𝑑𝑦 𝑑𝑥 . We defined the normal to a graph at a particular point as the line that is perpendicular to the tangent at that point . QUESTIONS: p #1, 2, 5cd, 6cd, 7bc, 11, 12

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