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Unformatted text preview: Math 006 (Lecture 17) The Derivative
dy Deﬁnition 1. For y = f (x), we deﬁne the derivative of f at x, denoted by f (x), or dx df , to be dx f (x + h) − f (x) f (x) = lim h→0 h if the limit exist. If f (x) exists for each x in the interval a < x < b, then f is said to be diﬀerentiable over a < x < b. Summary 1. Slope of the tangent line: f (x) is the slope of the line tangent to the graph of f at the point (x, f (x)). 2. Instantaneous rate of change: f (x) is the instantaneous rate of change of y = f (x) with respect to x. 3. Velocity: If f (x) is the position of a moving object at time x, then v = f (x) is the velocity of the object at that time. Procedure for ﬁnding the derivative Example 1. Find the derivative of f at x, for f (x) = 2x − x2 . Step 1: Find f (x + h) Step 2: Find f (x + h) − f (x) Step 3: Find f (x + h) − f (x) h Step 4: Find lim f (x + h) − f (x) h→0 h 1 Example 2. Find f (x) for each of the following functions: (a) f (x) = x3 (b) f (x) = (c) f (x) = 1 x √ x Basic Diﬀerentiation Properties
Theorem 1. (Power Rule) If y = f (x) = xn where n is a real number, then f (x) = nxn−1 . Example 3. Find f (x) for each of the following functions: (a) f (x) = 3 (b) f (x) = x5 (c) f (x) = x3/2 (d) f (x) = x−3 1 (e) f (x) = √ 3 x (f) f (x) = x 2 .
√ 2 Theorem 2. If y = f (x) = ku(x), where k is a constant, then f (x) = ku (x). If y = f (x) = u(x) ± v (x), then f (x) = u (x) ± v (x). Example 4. Find the derivative of (a) f (x) = 3x4 − 2x3 + x2 − 5x + 7 (b) f (x) = 3 − 5 x2 (c) y = 6v 4 − √ 5 v (d) d dx 1 3 x2 +√ − 5x4 2 x (e) f (x) = x2 + 25 x2 3 Example 5. An object moves along the y axis so that its position at time x is f (x) = x3 − 6x2 + 9x (a) Find the instantaneous velocity function v . (b) Find the velocity at x = 2 and x = 5. (c) Find the time(s) when the velocity is 0. Example 6. Let f (x) = 2x3 − 9x2 + 12x − 54. (a) Find f (x). (b) Find the equation of tangent to the graph of y = f (x) at x = 0. (c) Find the value(s) of x where the tangent to the graph of y = f (x) at x is horizontal. 4 ...
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 Fall '09
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 Math, Derivative

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