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**Calculus 2.1 Introduction to Differentiation**

Mrs. Kessler

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Rate of Change To begin our study of rates and changes we must realize that the rate of change is the change in y divided by the change in x. We are assuming two points With a straight line, this has been the slope, m. with However, with a curve, this is a little more complicated.

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**Examples of tangents of curves**

Since the curve is increasing and decreasing at different rates, we are looking for the instantaneous rate of change at a particular point. This means we need the slope of the tangent line at the point P.

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**Approximating the slope of the tangent line with the secant line**

x + x

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**What happens as x gets smaller and sMALLER?**

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**Definition of the Slope of a Graph**

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**Definition of Derivative**

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**Example 1 Find the slope of f(x) = 7x - 4 using the limit process**

But you already knew that.

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**Ex 2. Find f ′(x) if using the limit process**

Notice that the slope, or derivative, has a variable.

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**Ex. 3 Find the equation of the line tangent to at (1, 2)**

From before Let’s look graphically at this:

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**Ex. 3 Find the equation of the line tangent to at (1, 2)**

From before At x =1 m = ¼ point = (1, 2) y - y1= m (x – x1)

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**Find f′(x) if using the limit process**

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**Find f′(x) if using the limit process**

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**Find equation of the line tangent to and parallel to y = 2x -7**

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**That’s what we are looking for! Find its equation.**

Find equation of the line tangent to and parallel to y = 2x -7 Original function That’s what we are looking for! Find its equation. Original function plus line

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**Find equation of the line tangent to and parallel to y = 2x -7**

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**is the slope of the tangent line at any point x.**

Continued: is the slope of the tangent line at any point x. We want a tangent line parallel to y = 2x – 7 We can see m = 2 in y = 2x – 7 and from the calculus, m = . Now set the slopes equal and solve for x. Now sub back into the original function, and find y. y = 1/4

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**Now find the equation y – y1 =m ( x – x1)**

At the point (17/16, ¼), the tangent line is parallel to Now find the equation y – y1 =m ( x – x1)

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Day Questions?????????

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**y = x2. The two tangent lines intersect at (1, -3). **

Find their equations. (x1, y1) (1, -3) Let the red dot be the point (x1, ,y1). By the delta process the derivative is 2x. At the point (x1 ,y1), the derivative which is the slope = 2x1. Now use the slope formula setting the two =.

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**The tangent points are ( 3, 9) and (-1, 1) If x =3, f(x) = y = 9**

y = x2. The two tangent lines intersect at (1, -3). Find their equations. (x1, y1) Cross multiply and set equal to 0. Note: y1=(x1)2 (1, -3) The tangent points are ( 3, 9) and (-1, 1) If x =3, f(x) = y = 9 If x = -1, f(x) = y = 1

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**Now find the equation of the line.**

y = x2. The two tangent lines intersect at (1, -3). Find their equations. The tangent points are ( 3, 9) and (-1, 1) Now find the equation of the line. y - y1= m ( x – x1) (x1, y1) (1, -3) For first equation use ( 3, 9) and (1, -3) For second equation use ( -1, 1) and (1, -3) y = 6x – and y = - 2x -1 Graph and check.

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**Weird Tangents - no derivative**

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**Alternative form of the derivative**

provided the limit exists. exist and are equal.

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**Alternative form of the derivative**

at x = 4 Since the limit from left is not equal to the limit from the right, there is no limit, which means there is no derivative at x = 4 even though the function is continuous there.

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Continued

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**Using the TI 83/84 to draw the tangent line**

Enter the function into the y -editor. Graph the function: y = x sin(x) Suppose I want the tangent at x = π/3. Press 2nd Draw 5 Type π/3 Note: Decimal place was set a 3 fixed places.

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Clicker Question 1 What is the derivative function f '(x ) of the function ? (Hint: Algebra first, calculus second!) A. 12x 2 – (5/2)x -1/2 B. 12x 2 –

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