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MATH1141: Calculus I A4-1

TRU Open Learning

Assignment 4 (8%)

Assignment 4 is graded with a total of 100 marks and it contributes 8 percent

towards your course grade.

[10 marks] 1. If a ball is thrown vertically upward with a velocity of 80 ft/s,

then the height after t seconds is 80𝑡 − 16𝑡2.

(4 marks) a. What is the maximum height reached by the ball?

(6 marks) b. What is the velocity of the ball when it is 96 ft above the

ground on its way up? On its way down?

[8 marks] 2. A sample of a radioactive substance decayed to 78% of its

original amount after 3 years.

(4 marks) a. What is the half-life of the substance?

(4 marks) b. How long would it take the sample to decay to 40% of its

original amount?

[10 marks] 3. A kite 100 ft above the ground moves horizontally at a speed

of 8 ft/s. At what speed is the angle between the string and the

horizontal decreasing when 200 ft of the string have been let

out?

[8 marks] 4. At noon, ship A is 200 km east of ship B and ship A is sailing

north at 30 km/h. Ten minutes later, ship B starts to sail south at

35 km/h.

(4 marks) a. What is the distance between the two ships at 3 pm?

(4 marks) b. How fast (in km/h) are the ships moving apart at 3 pm?

A4-2 Assignment 4

[10 marks] 5. Find a linear approximation in each case and use it to

approximate the number. Give your answer as a fraction.

(5 marks) a. √39999

(5 marks) b. tan 46∘

[8 marks] 6. A child’s toy is a flat disk shaped as in the diagram, a square

with semi-circular ends. The side of the square is measured to

be 8 cm with an error of 0.05 cm. Use differentials to find the

percent error in the area.

[18 marks] 7. Find the absolute maximum and absolute minimum values of

f on the given interval.

(6 marks) a. 𝑓(𝑥) = 𝑥2+2𝑥+1

𝑥2+1 on [−1, 2]

(6 marks) b. 𝑓(𝑥) = 𝑥2𝑒2𝑥 on [−2, 1]

(6 marks) c. 𝑓(𝑥) = 𝑥 − 2cos𝑥 on [-2, 0]

[8 marks] 8. At 2 pm a car’s speedometer reads 30 mph. At 2:10 pm it

reads 50 mph. Show that at some time between 2 pm and

2:10 pm, the acceleration is exactly 120 miles per hour2. Identify

the theorem you have used.

MATH1141: Calculus I A4-3

TRU Open Learning

[8 marks] 9.

(4 marks) a. Sketch the graph of a function that satisfies all of the given

conditions. Note that you need to make ONE graph that

satisfies all the given conditions.

𝑓 ′(𝑥) > 0 if |𝑥| < 2, 𝑓 ′(𝑥) < 0 if |𝑥| > 2, 𝑓 ′(2) = 0,

lim 𝑥→∞

𝑓(𝑥) = 1 , 𝑓(−𝑥) = −𝑓(𝑥),

𝑓"(𝑥) < 0 if 0 < 𝑥 < 3, 𝑓" (𝑥) > 0 if 𝑥 > 3.

(4 marks) b. Find lim 𝑥→0

𝑥2−2𝑥

2𝑥+2 sin 𝑥 .

[12 marks] 10. For 𝑓(𝑥) = 𝑥𝑒−𝑥 2 ,

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

and where f is decreasing.

(3 marks) b. Give local maximum and local minimum values of f.

(4 marks) c. Give in interval notation, the intervals where f is concave

up and concave down. Give the coordinates of the point(s)

of inflection.