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Rates of Change. Objectives: Be able to find the average rate of change of an object. Be able to find the instantaneous rate of change of an object. Be able to find the speed of an object. Critical Vocabulary: Rate of change. I. Rates of Change. What is the derivative used for?.
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Rates of Change • Objectives: • Be able to find the average rate of change of an object. • Be able to find the instantaneous rate of change of an object. • Be able to find the speed of an object. Critical Vocabulary: Rate of change
I. Rates of Change What is the derivative used for? The derivative also can be used to determine the rate of change of one variable with respect to another. _____________ _____________ _____________ _____________ _____________ Position Function f(t) = blah blah blah ____________________________________________________________ ____________________________________________________________
I. Rates of Change As time changes, so does the position Distance ΔDistance Δs Rate of Change Time ΔTime Δt ______________________________________________ If a billiard ball is dropped from a height of 100 feet, its height s (in feet) at time t (in seconds) is given by the position function s(t) = -16t2 + 100. Find the average velocity over the time interval [1, 2]. A billiard ball is falling from ___ feet to ___ feet in 1 second at an average rate of ___ ft/sec.
II. Instantaneous Rates of Change f(x) = ___________________ f’(x) = ___________________ If a billiard ball is dropped from a height of 100 feet, its height s (in feet) at time t (in seconds) is given by the position function s(t) = -16t2 + 100. Find the instantaneous rate of change at 1 second and at 2 seconds. These rates represent the velocity of the object falling (implies direction).
III. Speed Speed is the absolute value of velocity. Speed cannot be negative. The position of a free falling object (neglecting air resistance) under the influence of gravity can be represented by the equation: s(t) = ½gt2 + vot + so t = _________________________________________ s(t) = _________________________________________ so = __________________________________________ vo = __________________________________________ g = __________________________________________ __________________________________________
III. Speed At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t) = -16t2 + 16t + 32 a. When does the diver hit the water? b. What is the divers velocity at impact?
