Topic: Table-­‐Chart Free Response Questions You will be given a table of values that represent the values of a function or its derivative. Understand that when you are given a rate you are given the derivative of a function. If this appears on the calculator active section: What you should know how to do: x(t) or s(t)
f (x)
Differentiation Ladder f '(x) For example: v(t)
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a(t)
f "(x)
Type I: One step down the ladder: Find the average rate of change of a function (one step down the ladder) by using average rate of change. o Example: If given velocity in table, find average acceleration. o Example: If given position, find average velocity. f (b) − f (a)
o
b−a
f (b) − f (a)
Type II: One step down the ladder: Use average rate of change (
) as an estimate for b−a
the derivative at a value not given on the table!!! v(2) − v(1) 18 − 15
o Example: To estimate the acceleration at t=1.5à à
= a(1.5) ≈ 3 2 −1
2 −1
v(t) 10 15 18 25 t 0 1 2 3 b
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b
∫a f (x)dx integral
1
Type III: Equal Position on the ladder: Use ( ) formula to f (x)dx or b−a
interval
b − a ∫a
estimate the average value. o Example: If given a table of temperature, find average temperature. o Example: If given a table of velocity, find the average velocity. b
1
o They often will give f (x)dx and ask you to explain its meaning in words. Make b − a ∫a
sure you say it is the average value of what is given in the table, give the units (same as table), and the time interval. Type IV: (One Step UP THE LADDER) Use Riemann sums (left, right, midpoint) or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). o The definite integral of a rate gives you the change in the parent (one step up). o Be ready for the initial value (anchor) problem. If you are given a table of rates, use a Riemann sum to approximate the definite integral and add that to the initial value. §
b
b
a
a
∫ f '(x)dx = f (b) − f (a) Therefore f (b) = f (a) + ∫ f '(x)dx b
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f (a) is given and you use a Riemann sum to find ∫ f '(x)dx a
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They could also give f (b) and ask for the initial value. •
Type V: FTC (One Step UP THE LADDER) b
o The will give you a table of the parent ( f (x) ) and ask for ∫ f '(x)dx . This is a FTC problem. a
b
∫ f '(x)dx = f (b) − f (a) . They also will ask you to explain its meaning in words. It is equal to a
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the change in f from t=a to t=b. Be sure to include units in your explanation. Use the chart to find f (b) and f (a) . Type VI: MVT (One Step Down) o They will ask if the derivative (of the function in the table) is ever equal to a given value f (b) − f (a)
over a time interval. If the average rate of change over the given interval is b−a
equal to the value and the function is differentiable (IMPORTANT TO STATE), then the MVT guarantees that f '(x) is equal to the average value. § Example: If L(t) is differentiable, is there a time c on 1 ≤ t ≤ 7 where L '(c) = 4 liters L(7) − L(1)
per second? Since = 4 and L(t) is differentiable, therefore the MVT 7 −1
guarantees a time t on 1 ≤ t ≤ 7 where L '(c) = 4 . L(t) Liters 4 9 28 t (time in sec) 1 3 7 Type VII: IVT (Equal Position on the ladder) o Did a given function ever achieve a value over a time interval? The IVT states that if a function is continuous (IMPORTANT TO STATE) and the function is equal to a value above and below the value in question, the IVT guarantees the function achieves that value. Find function value at the beginning and end of the time interval. One will often be above and one will be below. § Example #1. The continuous function x(t) has velocity v(t) as shown in the table below. For 0 < t < 60 , must there have been a time t when v(t) = 3 ? o Since v(0) = −10 < 3 < 15 = v(60) , and the x(t) is continuous, the IVT guarantees a time t on 0 < t < 60 where v(t) = 3 . § Example #2. What is the minimum number of times the particle must have stopped? x(t) is continuous, and since v(0) < 0 and v(20) > 0 , v(20) > 0 and v(40) < 0 , v(40) < 0 and v(60) > 0 ; the particle’s velocity must have been zero and therefore must have stopped at least three times. -­‐10 4 -­‐5 15 v(t) m/s t (time in sec) 0 20 40 60