CABChp4-6ANTs4 - CalcAB Chp 4-6A Related Rates Algebra and...

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CalcAB Chp 4-6A Related Rates, Algebra and Geometry Notes 1. Related-rate problems involve movement. The rate something changes or moves can be any value (slow, fast, positive, negative, zero). The RATE OF CHANGE of something for us means a DERIVATIVE is used. All derivatives in this section are done WITH RESPECT TO TIME; IMPLICIT DIFFERENTIATION is needed in order to accommodate things changing with respect to time and to accommodate more than one variable without substitution. There is a difference between optimization problems and rate problems. Optimization problems focus on finding specific values that maximize or minimize some quantity; the derivatives are set equal to zero. Whereas, related-rate problems are about determining a value of something at any time, not just when a situation is at an extreme value (max/min); derivatives are used because (instantaneous) rates are involved. 2. Ex(algebra) Find the rate y changes as x changes for each function. (a) Find dy/dt when x = 1 and dx/dt = 5 for y = 2x 3 + 3 (b) Find dy/dt when x = 3 and dx/dt = 6 for y x 2 = 81 3. Ex(algebra, point moves along a function path) A point is moving along the parabola 6y = x 2 . When x = 6, the abscissa ( x-term ) is increasing at a rate of 2 ft/sec; at what rate is the ordinate ( y-term ) changing at that instant? Organize given information; translate the words into mathematical notation. “x -term is increasing at a rate of 2 ft/sec translates to dx/dt = 2 ft/s rate the y-term is changing translates to dy/dt = ?? ft/s "when x = 6" is a value to use in the derivative after it has been determined. In the problem stated, the x value is changing with respect to time (seconds); the only way to solve how fast the y-value is changing is to work with the derivative implicitly, with respect to time (dt). After the derivative has been determined , any known values can be plugged in. Use x = 6 and dx/dt = 2 in the derivative to solve for the unknown dy/dt.
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CABChp4-6ANTs4 - CalcAB Chp 4-6A Related Rates Algebra and...

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