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MATH1141: Calculus I A5-1

TRU Open Learning

Assignment 5 (8%)

Assignment 5 is graded with a total of 100 marks and it contributes 8 percent

towards your course grade.

[12 marks] 1. For 𝑓(𝑥) = 𝑥6 − 3𝑥2

(3 marks) a. Give in interval notation the intervals where f is increasing

and where f is decreasing.

(3 marks) b. Give in interval notation the intervals where f is concave

up and where f is concave down.

(3 marks) c. Give the coordinates of any points of inflection.

(3 marks) d. Sketch the curve.

[12 marks] 2. Repeat question 1 above for 𝑓(𝑥) = 𝑒−𝑥 2

[10 marks] 3. A wire 10 m long is to be cut in two pieces. One piece is bent

into a square and the other into a circle. How should the wire

be cut so that the total area enclosed is

(5 marks) a. a maximum?

(5 marks) b. a minimum?

Give exact answers and then to one decimal place.

[10 marks] 4. A boat leaves a dock at 2:00 pm and travels south at a speed

of 20 km/h. Another boat has been heading east at 15 km/h and

reaches the same dock at 3:00 pm. At what time were the two

boats closest together?

A5-2 Assignment 5

TRU Open Learning

[10 marks] 5. Consider the cost function, 𝐶(𝑥) = 25000 + 100𝑥 + 0.1𝑥2

where C is given in dollars. The average cost is given by 𝑐(𝑥) = 𝐶(𝑥)

𝑥 .

(3 marks) a. Find the cost, average cost, and marginal cost at a

production level of 100 units.

(5 marks) b. Find the production level that will minimize the average

cost.

(2 marks) c. Find the minimum average cost.

[10 marks] 6. Use Newton’s method to approximate the root of 𝑥4 + 𝑥 −

4 = 0 in the interval [1,2] correct to six decimal places.

[10 marks] 7. Use Newton’s method to find the positive root of 3 sin 𝑥 = 𝑥

correct to six decimal places.

[8 marks] 8. Find f if 𝑓" (𝑡) = 𝑒2𝑡 − 4 cos 𝑡 if 𝑓(0) = 1

4 and 𝑓 (

𝜋

2 ) = 0

.

[8 marks] 9. Find a function f such that 𝑓 ′(𝑥) = 𝑥5 + 20 and the line

12𝑥 + 𝑦 = −20 is tangent to the graph of f.

[10 marks] 10. A car braked with constant deceleration of 5 m/s2 producing

skid marks measuring 60 m before coming to stop. How fast

was the car travelling when the brakes were first applied? Give

your answer in km/h.