Math - Difference Quotients and Average Instantaneous Rate of Change PDF

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Difference Quotients and Average Instantaneous Rate of Change...

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Learning Target: I can use difference quotients to determine average and instantaneous rates of change at a point and solve the selected problems with at least 80% accuracy. [2.1] Do Now: Today we are introduced to one of the most important concepts in calculus. NOTES: Defining Average and Instantaneous Rates of Change at a Point Rate of change (slope) refers to a rate that describes how one quantity changes in relation to another quantity. o What is the average rate of change of a function? 𝒇(𝒃)−𝒇(𝒂) ∆𝒚 Slope of secant line on an interval: ∆𝒙 = 𝒃−𝒂 For linear functions, the rate of change is constant, but what is the rate of change for non-linear functions? It changes!! Hence, we talk about what the average rate of change equals ON AN INTERVAL. o What is the instantaneous rate of change of a function? Slope of tangent line at a point. In geometry, we learned that a secant line is a line that intersects a circle, or some other curve, in two places. We also learned that a tangent line is a line that intersects a circle at one point. In our study of calculus, we need to clarify, classify, and truly understand these two big ideas. For general curves, we need a better definition. We will be concerned with the tangent line drawn at a point on a curve. Most of us believe that a tangent to a curve is a line that intersects the curve at exactly one point. However, in the figure below, you can see that the tangent line touches the curve at several points. So, a better way to describe a tangent line would be to say,

**This idea will help us find the slope of the curve at a point. Also, this slope of the curve at a specified point will be referred to as the slope of the __________ line, __________________ rate of change, or ____________. A) Finding the slope of the secant line through two points 𝑃(𝑎, 𝑓(𝑎)) and 𝑄(𝑏, 𝑓 (𝑏)) will give you the average rate of change over the interval [𝑎, 𝑏]. Average Rate of Change: AROC=

∆𝒚

∆𝒙

= 𝒎𝒔𝒆𝒄 =

B) Finding the slope of the tangent line at a point will give us the instantaneous rate of change at that point. However, we only have a single point, so using the average rate of change formula produces an indeterminate form.

𝒇(𝒃)−𝒇(𝒂) 𝒃−𝒂

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QUESTIONS: 1) How do we find the slope of a curve? The slope of a curve is always changing unlike a line, which has a constant slope. The slope of a curve changes from point to point. We can approximate the slope of a curve at a single point by finding the slope of the tangent line at that point! Approximating tangent lines makes use of a secant line through the point of tangency and a second point on the graph, as shown in the figure below. 2) What does the expression

𝒇(𝒙+𝒉)−𝒇(𝒙) (𝒙+𝒉)−𝒙

represent? What does this

expression simplify to?

3) As ℎ, the distance between the two 𝑥 -values, approaches zero, what happens to the secant line?

4) What does the limit 𝐥𝐢𝐦

𝒇(𝒙+𝒉)−𝒇(𝒙)

5) What does the limit 𝐥𝐢𝐦

𝒇(𝒙)−𝒇(𝒂)

𝒉

𝒉→𝟎

represent?

𝒙−𝒂 𝒙→𝒂 𝑓(𝑥+ℎ )−𝑓 (𝑥)

How does it relate to lim

ℎ→0

ℎ

represent?

?

TASK #1: The function, 𝑓(𝑥 ) = 𝑥 2 is graphed below. Find the average rate of change of 𝑓(𝑥 ) on [−1, 2] and the instantaneous rate of change at the point (2, 4).

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TASK #2: 𝑓(𝑥 ) is a continuous function having selected values of 𝑥 in the table below. Find the average rate of change on the closed interval [2, 4] and use the data in the table to approximate the instantaneous rate of change of 𝑓(𝑥 ) at 𝑥 1.

PRACTICE: 𝜋 1. For 𝑓(𝑥 ) = cos 𝑥, find the average rate of change on the closed interval [0, 2 ] and approximate the 𝜋

instantaneous rate of change of 𝑓(𝑥 ) at 𝑥 = . 4

2. Let ℎ (𝑥 ) be a differentiable function with selected values given in the table below. What is the average rate of change of ℎ (𝑥 ) over the closed interval 1 ≤ 𝑥 ≤ 7?

A. −

2

3

B.

1

6

C. 5

3. [CALCULATOR: DO NOT ROUND UNTIL THE VERY END!]

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4. Let 𝑝 be the function defined by 𝑝(𝑥 ) = 𝑒 2𝑥 sin 𝑥. Find the average rate of change of 𝑝 on the interval 3𝜋 𝑥≤ .

𝜋 2

≤

2

*5. The graph of the function 𝑓 and a table of selected values of 𝑓(𝑥 ) are shown above. The graph of 𝑓 has a horizontal tangent line at 𝑥 = 4, is concave down for 0 < 𝑥 < 6, and is linear for 𝑥 ≥ 6. (a) Approximate the value of 𝑓 ′ (5.5) using data from the table. Show the computations that lead to your answer. 𝑓(𝑥+ℎ )−𝑓 (𝑥) = 0? Give a reason for your answer. (b) Is there a value of 𝑥, for 0 < 𝑥 < 6, such that lim ℎ ℎ→0

(c) For each of the following limits, find the value or explain why it does not exist.

6. Read the following:

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