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## UNIT 1B LESSON 4

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**UNIT 1B LESSON 4**Slopes of Tangents**Slopes as Rate of Change**To begin our study of rates and changes we must realize that the rate of changeis thechange inydivided by thechange inx. With a straight line, this has been theslope of the secant, m. We are assuming two points where x has been increased by an amount h Q P**Approximating the slope of the tangent line with the secant**line x means the change in x We sometimes use h instead of x P x + x x**or ∆x→0**Tangent to a Curve**Examples of Tangents of Curves**Since the curves are increasing and decreasing at different rates, we are looking for the instantaneousrate of changeat a particular point. This means we need the slope of thetangent lineat the point P.**Tangent to a Curve**• The process becomes: • 1. Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. • 2. Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. • 3. Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.**EXAMPLE on Page 1 of Unit 1 Lesson #4The point P(1, – 5)**lies on the curve y = x2 – 6x. P(1, - 5) Find slope (m) at each point m = –5 –– 2.75 = – 4.5 1 – 0.5 m = –5 –– 4.59 = – 4.1 1 – 0.9 m = –5 –– 4.96 = – 4.0 1 – 0.99 - 4.0 m = –5 –– 5.04 = – 4.0 1 – 1.01 m = –5 –– 5.39 = – 3.9 1 – 1.1 Calculate values ofy with values of x very close to 1. m = –5 –– 6.75 = – 3.5 1 – 1.5**Find the slope of the tangent line to the curve y = x2 –**6x, at P(1, – 5) using limits. m = lim f (x + h) – f (x) h→0 (x + h) – x m = lim [(x + h)2 – 6(x + h)] – [x2 – 6x] h→0 (x + h) – x m = lim x2 + 2xh + h2 – 6x – 6h – x2 + 6x h→0 h m = lim h(2x + h – 6) h→0 h Slope at x = 1 is 2(1) – 6 = –4 m = 2x – 6**P(1, - 5)**Find the equation of the tangent line to the curve y= x2 – 6x. at P(1, – 5) From previous work Check this result on your calculator Enter Y= x2 – 6x Graph. DRAW 2nd PRGM 5:Tangent Type in the x-value where you want the tangent line drawn.