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ID: 2829827 • Letter: Q
Question
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For the graph below, sketch two functions F such that F' = f. In one case let F(0) = 0 and in the other, let F(0) = 1. Which of the following best matches your sketch? You are given the following graph of dP/dt. Use the figure above and the fact that P(0) = 12 to find the values below. Given the values of the derivative f'(x) in the table and that f(0) = 200, estimate the values below. Find the best estimates possible (average of the left and right hand sums). Assume f is given by the graph in the figure. Suppose f is continuous and that /(0) = 0. You are given the graph of f(x) shown below. Sketch two functions F where F '(x) = f(x). In one case, let F(0) = 0 and in the other, let F(0) = 1. Which of the following best matches your sketch?Explanation / Answer
1.
Slope of F(x) first decreases,becomes zero and then increases. So F(x) is of the form in graph C.
WHen F(0)=0, the graph of the curve is the one down.
WHen F(0)=1, the graph of the curve is the one up.
2.
dP/dt = the given curve as function of time f(t)
=> dP = f(t) dt, where f(t) is the curve plotted
integrating both sides for t=0 to t0
P(t0)-P(0) = area under the curve f(t) from t=0 to t0
P(1)-P(0) = -1 => P(1) = P(0)-1 = 11
P(2)-P(0) = -2 => P(2) = P(0)-1 = 10
P(3)-P(0) = -2.5 => P(3) = P(0)-1 = 9.5
P(4)-P(0) = -2 => P(4) = P(0)-1 = 10
P(5)-P(0) = -1 => P(5) = P(0)-1 = 11
3.
4.
As seen in question 2,
f(x)-f(0) = area under the curve of f'(t) between t=0 to x
f(3)-f(0) = 2*5 + 1*-5 => f(3) = f(0)+5=5
f(7)-f(0) = 2*5 + 1*-5 + 1*10 + 2*-10 + 1*5=>f(7)=f(0)+0 = 0
integral of f'(x) from x=0 to 7 = 2*5 + 1*-5 + 1*10 + 2*-10 + 1*5 = 0
Slope of f(x) between x=0 and 2 is +5, and the only curve satisfying this condition is the bottom right curve.
5.
f(x) is always positive. So, F(x) is always increasing. So the answer is C.