Advanced Mathematics

Q 21 Watch the video and then solve the problem given below. Click here to watch the video. The graph in the example has a local maximum of 2 that occurs at x = 1. What are the coordinates of the point?

(b) Show that for all n E N, if A₁, A2,..., An and B are any sets, then

20. Complete parts (a)-(d) for the functions f(x) = -x² +7 and g(x) = -2x +4.

Problem 1 A line from point A (E=1200.49', N=1800.98') to point B (E=2200.95', N=1900.53') intersects a circle of radius 550.00' that is centered on point C (E=1700.00', N=2100.00'). Draw a sketch. Show all calculations. (a) Check if line and circle intersect (calculation - "no" is not an option). (b) If they do intersect, calculate the coordinates of the two intersection points.

Problem 2 A line goes from point A (X=1800.00', Y=1900.00'), to point B (X=2100', Y=-2000.00' *). What are the coordinates of a point C that was established by turning a left angle a=65° from AB, and a right angle ß=46° from BA? Show all calculations. (* Note the minus)

Find the function that is finally graphed after the following transformations are applied to the graph of y=√x in the order listed. (1) Shift up 5 units (2) Reflect about the x-axis (3) Reflect about the y-axis

Question 1 a) The time rate of change of a falling object's height can be given by the differential equation where y = height of the object, a = acceleration and t = time. An object is dropped from a height of 500m, with an acceleration due to gravity of -9.8m/sec², (i)Express the height of the object as a function of time. (ii) When will the object reach ground level? b) The acceleration (a =) of a car can be shown to obey the following linear differential equation = 0.05(250-v), where v is the velocity of the car. (i) Express the velocity of the car as a function of time. (ii) How long does it take the car to accelerate from rest to 200km/h? c) Find the general solution of d'r dr ds² ds - 2r=0 d) Find the first four non-zero terms of the power series solution for the differential equation y" + x²y = 0.

Consider the function f(x)=2/x. The goal is to compute the first derivative of the function at = 2 numerically using the following approximations with different grid resolutions (Step sizes) and comparing the values to the exact solution.

a) Use double integrals to find the volume of the solid lying under the surface z = 1+x+y and above the region in the xy-plane bounded by x = y² and .x = 4. b) Find the mass of the region bounded by the curve r = 2 sin given that the density across the region is constant. c) An ice cream cone can be described as being a solid bounded below by the cone ==√√3(x² + y²) and above by the hemisphere .x² + y² +2²=1, z 20. The volume of such a cone (ice cream and cone combined) can be calculated using triple integrals. (i) Set up the triple integral in rectangular co-ordinates to calculate the volume. (ii) Set up this triple integral in cylindrical co-ordinates to calculate the volume. (iii) Set up this triple integral in spherical co-ordinates to calculate the volume. (It may be helpful to know that the cone makes an angle of with the z axis.) (iv)Calculate the volume of this ice cream cone.

Assume a continuous function f(x) defined on x axis with a uniform grid of spacing h. Using appropriate Taylor series expansions2, find the leading order