Calculus Answers

Suppose a person standing at the top of a building 112 ft high throws a ball vertically upward
with an initial velocity of 96 ft/sec.
(a) Find the velocity of the ball at time

Find T,N,B if possible for the following curves
(a)r(t)=(t³/3)i+(t²/2)j,t>0
(b) r(t)=(t³/3)I+(t²/2)j,t>=0

Prove or disprove that the following vector function is discontinuos
r(t)=(cost)i+(sint)j+[t] k

Prove that if y= √x + 1/4 sin(2x^2)
dy/dx= (1/2√x) + 2xcos(2x)^2

An automobiles velocity starting from rest is where v is measured in feet per second. Find
i) Its velocity after 5 seconds
ii) Its average velocity during the 10th second
iii) Its acceleration after 2 seconds
iv) Its average acceleration from t=2 to t=4

a) If the saving function is given by S° = 0.03Y^2 - 2Y + 150
Calculate the values of marginal propensity to save (MPS) and marginal propensity to
consume (MPC) when Y=40.

b) Find the second derivatives of these functions:
i) y=5x^2+350x-120 [3 marks]

ii) y= ln( x^2+4) [3 marks]

c) Find the relative maximum and minimum fory=3x^2+30x+605. [4 marks]
d) Find the absolute maximum and minimum for y=x^2-2x+3 between x=0 and x=3. [4 marks]

From a box containing 4 black balls, 2 green balls, and 1 white ball, 3 balls is drawn in succession, each ball is not replaced in the box before the next draw is made. If X is the number of green balls drawn.

(a) What are the values for the set of x?
(b) What is the probability of having 2 green balls? (Fraction form)
(c) Make a probability distribution table.

A certain factory manufactures rectangular papers. Various sizes are produced by increasing the length by 6 centimeters each second and by decreasing its width by 5 centimeters each second. At a certain instant, the length of the paper is 3 centimeters, and the width is 8 centimeters. What is the rate of change of the area of the paper at that instant?

Bentwood boxes are made out of one piece of wood that is steamed and bent to form a box (tops and bottom of the box are made separately). 

Suppose you have a cedar plank of length 36 inches and width 3 inches. For the purpose of this problem, suppose the thickness of wood is negligible.
* (a) If you use this plank for the sides of the box only, find the dimensions of the largest (in terms of volume) rectangular bentwood box that you can make. 

* (b) Would your answer to part a) change if the width of the plank was different? Explain why or why not. 

* (c) Suppose now that you want to use this plank to make the sides and the bottom of the box. You are only allowed to make one cut to cut off a piece of the plank for the bottom of the box; you will then bend the rest of the plank to form the sides of the box. Find the dimensions of the largest (in terms of volume) bentwood box that you can make. 


Let f(x) = |2x − 10| + 2 .
a) Show that f is continuous at x = 5
b) Show that f is not differentiable at x = 5