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1 2.7 – Tangents, Velocity, and Other Rates of Change

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2 Definition: Secant and Tangent Lines f(x2)f(x2) f(x1)f(x1) x1x1 x2x2 P Q Secant Line – A line passing thorough two points on a graph of a function. P f(x1)f(x1) x1x1 f(x2)f(x2) Q x2x2 f(x2)f(x2) Q x2x2 f(x2)f(x2) Q x2x2 Tangent Line – A line that touches the graph at a point. The tangent line may cross the graph at other points depending on the graph.

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3 Slope of the Tangent Line The slope of the tangent line though (x 1, f(x 1 )) and (x 2, f(x 2 )) can be found using the slope formula. The slope of the secant line will be:

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4 Slope of the Tangent Line Now let x 1 = a and x 2 = x. This changes the slope of the secant line formula to: f(x)f(x) f(a)f(a) a x P Q

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5 Slope of the Tangent Line From previous lessons, we learned that we can estimate the slope of a tangent line x = a selecting a second point x that is very close to a. In other words, we will let x a. Therefore, Note: This formula is more useful when finding an actual numeric value for the slope of a tangent line.

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6 Slope of the Tangent Line Now let x 1 = a and x 2 = a+h, where h is the distance from x 1 to x 2. This changes the formula to: f(a+h) f(a)f(a) a a + h P Q f(x2)f(x2) f(x1)f(x1) x1x1 x2x2 P Q

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7 Slope of the Tangent Line To determine the slope of the tangent line using this form, let a + h approach a. This is equivalent to allowing h → 0. f(a+h) f(a)f(a) a a + h P Q ← means h → 0 Note: This formula is more useful when finding an equation (formula) for the slope of a tangent line.

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8 The Derivative The derivative of a function f at a number a, denoted by f ′(a), is

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9 Interpretations of the Derivative The tangent line to y = f (x) at (a, f (a)) is the line through (a, f (a)) whose slope is equal to f ′(a), the derivative of f at a. The derivative f ′ (a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

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10 The Derivative 4. A particle moves along a straight line with equation of motion s(t) = t -1 – t, where s is measured in meters and t in seconds. Find the velocity and speed when t = 5.

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11 The Derivative 5.If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, it’s height (in meters) after t seconds is given by h(t) = 10t – 1.86t 2. (a) Find the velocity of the rock after one second. (b) Find the velocity fo the rock when t = a. (c) When will the rock hit the surface? (d) With what velocity will the rock hit the surface?

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12 The Derivative 6.The number of bacteria after t hours in a controlled laboratory experiment is n = f(t). (a)What is the meaning of the derivative f '(5)? What are it’s units? (b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, f '(5) or f '(10)? If the supply of nutrients is limited, would that affect your conclusion? Explain.

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