Calculus Answers

Laplace Transform

Question 1:
For the following transfer function:
H(s)= (s^2 -1)/(s^2+s+1)
Plot the poles and zeros in the s-plane and determine and sketch the inverse Laplace transform h(t ) using Matlab.

Question 2:

For the following transfer function:
H(s) = 3 + ( (s+4) / (s^2+3s+2) )
Find and sketch the inverse Laplace transform h(t) using Matlab..

Question 3:
Solve the following second order differential equation:
y\\ + 4y\ + 3y = 4 e^-3x , given that y(0) = 1 and y\ (0) = -1
and sketch the output y(t) using Matlab.

I-ANALYTIC GEOMETRY/RECTILINEAR MOTION
1.Find the tangent line as directed to the curve Y = X4 + 2X3 – 2X2 -3X + 3 perpendicular to the line: X – 3Y = 2.
2.Find the tangent of the line as directed to the curve Y = X4 + 4X3 – 8X2 + 3X + 70 with slope 3.
3.Find the equation of the line tangent to the curve Y = 3X2 – 4X and parallel to the line X – 2Y + 6 = 0.
4.Find the equation of each normal line to the curve Y= X3 – 4X that is parallel to the line X + 8Y -8 = 0.
5.A particle moves along a straight line according to the law: S = 132 + 10t – 6t2 + 3t3. Find: a.) velocity and acceleration at any time t? b.) Velocity at t = 2 and c.) Acceleration at t= 3.
II-MAXIMA AND MINIMA
1.A box with a square base is to have an open top. The area of the material in the box is to be 100 in square. What should the dimensions be in order to make the volume as large as possible?

Two roads intersect at point P at an angle of 120◦
, as shown in the figure. Car X is driving
from P towards A, and car Y is driving from P towards B. At a particular time, car X is 10
kilometers from P and traveling at 60 km/hr, while car Y is 12 kilometers from P and traveling
at 80 km/hr. How fast is the distance between the two cars changing?

I have the same question only with C(x)=72,000+40x and p(x)=300-x/20, 0≤x≤6000
(A) Find the maximum revenue.
​(B) Find the maximum​ profit, the production level that will realize the maximum​ profit, and the price the company should charge for each television set.
​(C) If the government decides to tax the company ​$5 for each set it​ produces, how many sets should the company manufacture each month to maximize its​ profit? What is the maximum​ profit? What should the company charge for each​ set?

2. For the function f(x) = 2x -arcsin x, a) locate all the local extrema and classify them
as maxima or minima. b) Are these local extrema also global extrema? Explain your answer.
4. For the curve y = ln(1+x^2), find a) intervals of increase and decrease and local extrema,
b) intervals of concavity and inection points, then use this information to c) sketch the graph.

3. Suppose f is an odd function which is differentiable everywhere and a is a positive number.
Show that for every a there exists c belong ( a; a) such that f'(c) = f(a)=a.

14. Two roads intersect at point P at an angle of 120, as shown in the figure. Car X is driving
from P towards A, and car Y is driving from P towards B. At a particular time, car X is 10
kilometers from P and traveling at 60 km/hr, while car Y is 12 kilometers from P and traveling
at 80 km/hr. How fast is the distance between the two cars changing?

17. You are designing a rectangular poster which will have a rectangular printed area surrounded by margins of width 10 cm at the top and the bottom, and margins of width 7.5 cm at both sides. If the total area of the poster is to be 1m^2, find the maximum possible printed area.

1.Suppose that for all x >= 0, the average value of f(x) on [0; x] is equal to x. What is the
funtion f(x)?
2.The equation x^4+x^3-4 = 0 has a solution near x = 1. Use Newton's method to find this
solution correct to 8 signficant figures.
3.At a certain moment, a triangle has base length 10 cm and perpendicular height 8 cm. The
length is increasing at a rate of 1 cm/s while the height is decreasing at 1 cm/s. How fast is the
area changing? Is the area increasing or decreasing?

A man is walking along a sidewalk at the rate of 5 feet/sec. A searchlight on the ground 30 feet from the walk is kept trained on him. At what rate is the searchlight revolving when the man is 20 feet away from the point on the sidewalk nearest the light?