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

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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.

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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.

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

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

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6 Definition of the Slope of a Graph

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7 Definition of Derivative

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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.

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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.

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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:

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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 )

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

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

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

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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.

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

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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

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

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

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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 )

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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

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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.

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23 Weird Tangents - no derivative

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

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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.

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26 Continued

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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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