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Question

Q1)

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Q2)

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Q3)

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Q4)

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Q5)

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Q6)

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Q7)

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Q8)

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Q9)

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Q10)

Sketch two functions F such that F' = f. In one case let f(0) = 0 and in the other, let f(0) = 2 Use the figure below and the fact that P = 9 when t = 0 to find values of P when t = 1, 2, 3, 4 and 5. Estimate f(x) for x = 2, 4, 6, using the given values of f'(x) and the fact that f(0) = 105 Assume f' is given by the graph in the figure. Suppose f is continuous and that f(0) = 0. Find all x with f(x) = 0. (Enter your answers as a comma-separated list) Sketch a graph of f over the interval 0

Explanation / Answer

Q1) 4th graph since slope is decreasing and increasing

Q2) at t = 1,2

dP/dt = -1

P(t) - P(0) = -t

P(t) = P(0) - t

at t = 1, P(1) = 9-1 = 8

at t = 2, P(2) = 9-2 = 7

at t = 3, dP/dt = t - 3

P(3) = P(2) + 3^2/2 - 3*3 - 2^2/2 + 3*2 = 7 + 3^2/2 - 3*3 - 2^2/2 + 3*2 = 6.5

P(4) = P(3) + 4^2/2 - 3*4 - 3^2/2 + 3*3 = 7

P(5) = P(4) + 5 - 4 = 7+1 = 8

Q3) y = mx + c

f(x) = f'(x)*x + c

f(x) = 105 = c

f(2) = 17*2 + 105 = 139

f(4) = 22*4 + 105 = 193

f(6) = 28*6 + 105 = 273

Q4) f(2) = 2*2+0 = 4

f(3) - f(2) = -2*(3-2) = -2

f(3) = 2

f(4) = f(3) + 4*(4-3) = 6

f(6) = f(4) - 4*2 = -2

f(7) = -2 + 2*1 = 0

Option D is the correct graph

x = 5.5 and 7 when f(x) = 0

Q5) till 15 slope is negative, hence must be A or B graph

since slope is not constant from 15 to 40, g(x) should be concave or convex, hence GRAPH B is the correct one

Critical points at x = 15 and 40

Inflection points at x = 10 and 20

Q6) Graph A is correct

x1 is local minima

x2 is inflection point

x3 is inflection point

Q7) a) moving upward

not moving

moving downward

not moving

b) A and C

Q8) Reaches max when derivative of antiderivative is 0

2sin(x^2) = 0, x = 0 or sqrt(pi) or sqrt(2pi)

4cos(x^2) must be negative,

that means x = sqrt(pi) = 1.772

Q9) a) x = x5

b) x = x1

c) x = x3

d) x = x1

e) x = x1

f) x = x5

Q10) a) since given graph is antiderivative,

find slopes to find values of g(x)

G(6) = 4

b) (12,18)

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