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Clicker Question 1 What is the derivative of f (x ) = x 3 e 4x ? (Hint: e 4x = (e 4 ) x ) A. 3x 2 e 4x B. e 4x (x 3 + 3x 2 ) C. e 4x (4x 3 + 3x 2 ) D. x 3 + e 4x E. 12x 2 e 4x

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Clicker Question 2 What is the instantaneous rate of change of g (x ) = x /(x – 5) at x = 2? A. 1 B. -5/9 C. -5/(x – 5) 2 D. 5/9 E. 5/(x – 5) 2

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Derivatives of Trig Functions (3/4/09) It is easy to see by sketching that the derivative of the sine function looks an awful lot like the cosine function and the derivative of the cosine function an awful lot like the negative sine function. This is in fact the case. The proof of the first is similar to the case of exponential functions, but requires use of the sin(A + B) formula.

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Some old trig formulas Sine of a sum: sin(A+B) = sin(A)cos(B) + sin(B)cos(A) And to get the derivative of the cosine, we need: Cosine of a sum: cos(A+B) = cos(A)cos(B) – sin(A)sin(B)

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Working out the Sine Upon working through the definition of the derivative of the f (t) = sin(t), we get f '(t) = sin(t) * (the derivative of cos at 0) + cos(t) * (the derivative of sin at 0) Those two numbers are, respectively, 0 and 1, and so we are done. Note the similarity to the derivative of a x.

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Clicker Question 3 What is the slope of the tangent line to the curve y = sin(x ) at the point ( /6, ½)? A. 3 / 2 B. 1 / 2 C. cos(x ) D. 1 E. /6

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An aside: Another notation for f '(x) The notation d/dx is often used to denote the derivative of a function of x. If the function is denoted by y, we write dy/dx, as opposed to y '. The top tells you the name of the output, the bottom the input variable, and the d’s denote “change in.” Example: d/dx (x 4 ) = 4x 3

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Derivative of the Tangent Since the tangent function is defined as a quotient, guess what rule we can use to compute its derivative? We see that d(tan(t ))/dt = 1/cos 2 (t ) = sec 2 (t ) Note that it makes sense that this derivative function is always positive. (Why?)

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Assignment for Friday Read Section 3.3 and do Exercises 1-17 odd, 21, 23, 25, 33 and 37. Correct your test! Test corrections (for some) are due Tuesday (3/17) at 4:45 pm. Persons with score below 80 can correct their mistakes (on a separate sheet!) and get back 1/3 of the lost points (to a max of 80). Work to be done on own or help from me only – no collarboration.

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