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1 Calculus 2.1 Introduction to Differentiation Calculus 2.1 Introduction to Differentiation Mrs. Kessler.

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Presentation on theme: "1 Calculus 2.1 Introduction to Differentiation Calculus 2.1 Introduction to Differentiation Mrs. Kessler."— Presentation transcript:

1 1 Calculus 2.1 Introduction to Differentiation Calculus 2.1 Introduction to Differentiation Mrs. Kessler

2 2 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.

3 3 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.

4 4 Approximating the slope of the tangent line with the secant line P x x +  x

5 5 What happens as  x gets smaller and s M A L L E R ?

6 6 Definition of the Slope of a Graph

7 7 Definition of Derivative

8 8 Example 1 Find the slope of f(x) = 7x - 4 Example 1 Find the slope of f(x) = 7x - 4 using the limit process But you already knew that.

9 9 Ex 2. Find f ′(x) if using the limit process Ex 2. Find f ′(x) if using the limit process Notice that the slope, or derivative, has a variable.

10 10 Ex. 3 Find the equation of the line tangent to at (1, 2) Ex. 3 Find the equation of the line tangent to at (1, 2) From before Let’s look graphically at this:

11 11 Ex. 3 Find the equation of the line tangent to at (1, 2) Ex. 3 Find the equation of the line tangent to at (1, 2) From before At x =1 m = ¼ point = (1, 2) y - y 1 = m (x – x 1 )

12 12 Find f′(x) if using the limit process

13 13 Find f′(x) if using the limit process

14 14 Find equation of the line tangent to and parallel to y = 2x -7

15 15 Original function Original function plus line Find equation of the line tangent to and parallel to y = 2x -7 That’s what we are looking for! Find its equation.

16 16 Find equation of the line tangent to and parallel to y = 2x -7

17 17 We want a tangent line parallel to y = 2x – 7 We can see m = 2 in y = 2x – 7 and from the calculus, m =. is the slope of the tangent line at any point x. Continued: Now sub back into the original function, and find y. Now set the slopes equal and solve for x. y = 1/4

18 18 Now find the equation y – y 1 =m ( x – x 1 ) At the point (17/16, ¼), the tangent line is parallel to

19 19 Day Questions?????????

20 20 y = x 2. The two tangent lines intersect at (1, -3). Find their equations. Let the red dot be the point (x 1,,y 1 ). By the delta process the derivative is 2x. At the point (x 1,y 1 ), the derivative which is the slope = 2x 1. Now use the slope formula setting the two =. (1, -3) (x 1, y 1 )

21 21 y = x 2. The two tangent lines intersect at (1, -3). Find their equations Find their equations. (1, -3) (x 1, y 1 ) If x =3, f(x) = y = 9 If x = -1, f(x) = y = 1 The tangent points are ( 3, 9) and (-1, 1) Cross multiply and set equal to 0. Note: y 1 =(x 1 ) 2

22 22 y = x 2. The two tangent lines intersect at (1, -3). Find their equations. (1, -3) (x 1, y 1 ) The tangent points are ( 3, 9) and (-1, 1) Now find the equation of the line. y - y 1 = m ( x – x 1 ) For first equation use ( 3, 9) and (1, -3) For second equation use ( -1, 1) and (1, -3) y = 6x – 9 and y = - 2x -1 Graph and check.

23 23 Weird Tangents - no derivative

24 24 Alternative form of the derivative provided the limit exists. exist and are equal.

25 25 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.

26 26 Continued

27 27

28 28 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 2 nd Draw 5 Type π/3 –N–Note: Decimal place was set a 3 fixed places.


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