Calculus Unit 2 Test

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Name: APPLICATIONS OF DERIVATIVES UNIT 2 TEST A

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PART A : Knowledge/Understanding Show all appropriate calculations for full marks. 1. Given s(t) - -t3 +4t2 -10 describes the position of an object that moves along a straight line (s in metres, f in seconds) /? a) Determine the initial velocity of the object. [ > 13]

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b) Determine whether the object is moving towards or away from its starting point att = 2.

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d) Is the object speeding up or slowing down at t = 3?

2. Determine the maximum and minimum values of the following function in the given domain.

f(x]~2x+—;

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PART B: APPLICATION /THINKING FULL CALCULUS SOLUTIONS REQUIRED.

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A motorist starts braking for a stop sign and "t" seconds after braking began, the distance, in metres, from the front of the car to the sign is s(t) = 62-16t+t2 Does the car go beyond the sign before stopping?

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4. A particle is moving along a horizontal line according to the function 3 2 s (t) = -t + 2 It - 99t +18; 0 < / < 15 (s in metres, t in seconds). a) Determine velocity and acceleration functions then set up an interval chart to help analyze the motion of the particle. [ / £} /10]

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b) When and where does the particle change direction?

c) When is the particle moving in a positive direction?

d) When is the particle slowing down?

5. A manufacturing company finds that the profit in dollars for a production level of x hundred widgits per hour is p = 40oVl2:c - x2 -100 for 0<x<12 . a) What was the average rate of change of profit for a production level between 200 and 300 widgits in an hour? (to the nearest cent per widgit)

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b) Determine the rate of change of profit when 200 widgits are produced in an hour. (to the nearest cent per widgit) i-

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c) Will company profits ever stop increasing? If so when. If not, why not?

p CO) - -100

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d) The company promises to give out bonuses to their employees if their profits exceed $2500 per hour. Wi!l this ever happen? If so what would the production level need to be?

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6. A variety store buys chocolate bars for 35 cents and sells them for $1. The store sells 500 chocolate bars a week. For each cent that the store reduces the price it finds that it sells 20 more chocolate bars each week. For what price should the store sell the chocolate bars for a maximum profit?

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7. A box has square ends and the sides are congruent rectangles. The total area of the four sides and two ends is 96 square centimetres. What are the dimensions of the box if the volume is a maximum and what is the maximum volume?

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