If f is continuous and integral_0^4 f(x) dx = 10, find integral_0^2 f(2x) dx a.)

Question

If f is continuous and integral_0^4 f(x) dx = 10, find integral_0^2 f(2x) dx a.) integral_0^2 f(2x) dx = 10 b.) integral_0^2 f(2x) dx = 20 c.) integral_0^2 f(2x) dx = 5 d.) integral_0^2 f(2x) dx = 8 A graph of g(t) is given below. Let G(t) be an anti-derivative of g(t) satisfying G(0) = 5. Find the following: i.) each critical point of G(t) with its coordinates. a.) (0, 0), (2, 21), (4, 13), (5.15) b.) (0, 5), (2, 16), (4, 13), (5, 14) c.) (0, 0), (1, 13), (3, 17), (9/2, 14) d.) (0, 5), (2, 21), (4, 13), (5, 15) ii.) the points of inflection of G(t) a.) (1, 13), (3, 17), (9/2, 14) b.) (0, 0), (1, 13), (3, 17), (9/2, 14) c.) (2, 13), (3, 17), (5, 14) d.) (0, 0), (2, 13), (4, 17), (9/2, 14)

Explanation / Answer

6.

(i)

Since G(t) is the antiderivative of g(t) which is given in the graph the first derivative of G(t) will be g(t).Hence the critical points will be those points which are zeroes or roots of the given curve. Hence the critical points will lie at t=0,2,4,5

We have

G(0) = 5

G(2) = 5+16 - 21

G(3) = 5+16-8 = 13

G(4) = 5+16-8+2 = 15

Hence the critical points would be,

(0,5),(2,21),(4,13),(5,15)

Therefore the option d. is correct.

ii) The inflection point of G(t) will be where the second derivative of G(t) is zero, or in other words the first derivative of g(t) is zero. Therefore the inflection points willl lie where the slope of the curve is zero. The slope of the cruve is zero at t=1,3,9/2.

Hence the option a. is correct.

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