Calculus 1 (G8-Wed) HW Assignment #3 2023
Due date: 11:20 AM, March 01, 2023. In-class submission. Instructions:
Submit the first 10 (ten) questions only: Q1-Q10. The solutions must be in the
order as listed, that is, Q1, Q2, Q3, ..., Q10. (You can temporarily leave a blank for a
question without a solution and can go back to resolve it later.)
Submit the solution papers in the class on March 01, 2023. For any late submissions,
please submit your papers in my P.O. mailbox in front of O2.610 (outside the office and nearby the office door). Exercises for Chapter 3 1. (Growth of a tumor)
When the diameter of a spherical tumor is 8 mm, it is growing at a rate of 0.16 mm a
day. How fast is the volume of the tumor changing at that time?
2. A 6-ft man walks away from a 15-ft lamppost at a speed of 3 ft/s. Find the rate at which
his shadow is increasing in length.
3. Suppose that the demand equation for a monopolist is p = 100 − 0.01x and the cost
function is C(x) = 50x + 10, 000. (a) Find the marginal cost.
(b) Find the value of x that maximizes the profit and determine the corresponding price
and total profit for this level of production.
(Answer: Produce 2500 units and sell them at 75 per unit. The profit will be 52,500).
4. Given f (x) = x3 − 12x + 1
(a) Find the intervals on which f (x) is increasing or decreasing.
(b) Find the local maximum and minimum values of f .
(c) Find the intervals of concavity and the inflection points. (d) Sketch the graph of f (x). 1 Calculus 1 (G8-Wed) HW Assignment #3 2023
5. A box has a square base of side x and height y.
(a) Find the dimensions x, y for which the volume is 12 and the surface area is as small as possible.
(b) Find the dimensions for which the surface area is 20 and the volume is as large as possible.
Hint: (a) The volume V = x2y = 12. Minimize the surface area A = 2x2 + 4xy.
6. (a) Given f (x) = x (x2 − 1) (x2 − 3), prove that the equation f′(x) = 0 has four distinct real roots.
Hint: Use the Rolle’s Theorem.
(b) Use the Mean Value Theorem to prove the inequality |sin a − sin b| ⩽ |a − b| for all a and b.
(c*) Show that x > sin x, ∀x > 0.
Hint: Let f (x) = x − sin x. Can use MVT for f(x) and f(0), x > 0. [Or, alternatively,
can show that f ′(x) > 0 on (0, π) and f (0) = 0.]
7. Using the l’Hospital’s rule to find the limits e3x − 1 ln(ln x)  x 1  ex − 1 − x − 1 x2 (a) lim , (b) lim , (c) lim − , (d) lim 2 x→0 ln(x + 1) x→∞ x x→1+ x − 1 ln x x→0 x3 8. (The l’Hospital’s rule.
(a) Find the limit lim x ln 1 + 1 , and then show that lim 1 + 1 x = e. x x x →∞ x→∞
Application: Suppose an amount of 1 is invested in the bank, with continuous com-
pounding of interest at interest rate 100% (r = 1), find the the amount after 1 year. (b) Find lim 1 + 43x, (c) lim x ln x, (d) lim xx. x x →∞ x→0+ x→0+
9. (a) Use Newton’s method to find the root of the equation x5 − x4 + 3x2 − 3x − 2 = 0 in
the interval [1,2] correct to six decimal places.
(b) Use Newton’s method to find the root of the equation 3x4 − 8x3 +2 = 0 in the interval
[2,3] correct to six decimal places.
10. A particle is moving in a straight line with the given data. Find the position of the particle. 1 (a) v (t) = 3t − , s (0) = 1. 1 + t2
(b) a (t) = 2 sin t + 3 cos t, s (0) = 0, v (0) = 2.
The below questions are for additional practicing. They are NOT for Assign- ment #3 submission:
11. A paper cup has the shape of a cone with height 12 cm and radius 4 cm (at the top).
If water is poured into the cup at a rate of 2.5 cm3/s, how fast is the water level rising 2 Calculus 1 (G8-Wed) HW Assignment #3 2023 when the water is 6 cm deep?
12. A boat is pulled into a dock by a rope attached to the bow of the boat and passing
through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is
pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?
13. An earth satellite moves in a path that can be described by x2 y2 + = 1 72.5 71.5
where x and y are in thousands of kilometres. If dx/dt = 12900 km/h for x = 3200 km and y > 0, find dy/dt.
14. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is
sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?
15. The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment
when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate
of 0.2 m/s. How long is the ladder?
16. Between 0oC and 30oC, the volume V (in cubic centimeters) of 1 kg of water at a tem-
perature is given approximately by the formula
V = 999.87 − 0.06426T + 0.0085043T 2 − 0.0000679T 3
Find the temperature at which water has its maximum density.
17. (a) Find the extreme values on [−1, 4] of p (x) = xe−x2/8.
(b) Find the extreme values of f (x) = x3 ln x, for x > 0.
18. The total cost, in dollars, of producing x units of a certain product is given by x3 C (x) = 8x + 20 + 100
and the average cost is defined by A(x) = C(x)/x. (a) Find A′(x) and C′(x).
(b) Find the minimum of A(x) and the value x0 at which it occurs.
(c) Compare A(x0) and C′(x0). 3 Calculus 1 (G8-Wed) HW Assignment #3 2023
19. A model used for the yield of an agricultural crop as a function of the nitrogen level in
the soil (measured in appropriate units) is kN Y = 1 + N2
where k is a positive constant. What nitrogen level gives the best yield?
20. A farmer wants to fence an area of 2 million square feet in a rectangular field and then
divide it into three equal parts with two fences parallel to one of the sides of the rectangle
(see the below figure). Find the dimensions of the rectangular field to minimize the cost of the fence?
21. Cowboy Clint wants to build a dirt road from his ranch to the highway so that he can
drive to the city in the shortest amount of time. The perpendicular distance from the
ranch to the highway is 4 miles, and the city is located 9 miles down the highway. Where
should Clint join the dirt road to the highway if the speed limit is 20 mph on the dirt
road and 55 mph on the highway?
Hint: Minimize the total number of hours for the trip, which is √16 + x2 9 − x T (x) = + 20 55 Answer: x = 16 √ ≈ 1.56 (miles). 105
22. A rectangular storage container with an open top is to have a volume of 10 m3. The length
of its base is twice the width. Material for the base costs 10 per square meter. Material
for the sides costs 6 per square meter. Find the cost of materials for the cheapest such container. 4 Calculus 1 (G8-Wed) HW Assignment #3 2023
23. A company estimates that the marginal cost (in dollars per item) of producing x items is
1.92 − 0.002x. If the cost of producing one item is 562, find the cost of producing 100 items.
24. A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant
deceleration of 22 ft/s2. What is the distance traveled before the car comes to a stop?
25. A car is traveling at 100 km/h when the driver sees an accident 80 m ahead and slams
on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup? —END— 5