Task 1
Calculate the following integral. You must show the method you use.
Hint: The answer is in the form . You can derive to find a and
b. An alternative way is to do partial integration twice.
(d) The functions and intersect in two points.
Determine the points of intersection. The functions define an area in the xy plane. Determine the area of this area. Give the answers exactly.
Task 2
Function is called the Gaussian or normal distribution, and is the most
important function in statistics and probability. It describes the distribution of uncertainty in measurements with many independent error sources. (More precisely, f(x) given above is the standard normal distribution which has mean 0 and standard deviation 1.) Google -Standard Normal Distribution- to see an image of this.
a) Find T2 (x), the Taylor polynomial of degree 2, for f(x) if x = 0 (a = 0 in the formula).
Calculate T2(0.1) and T2(1). Compare with f(0.1) and f(1).
b) The probability that a measurement is between plus and minus one standard deviation is about. 68%. This is calculated by the integral
This integral cannot be expressed by ordinary functions that you know, and must be calculated in
other ways. Calculate and compare with it correct result
c) Fourth degree Taylor polynomial for Use this to
find a better approach to
Comment: From the Taylor series one can find an infinite series for I and other corresponding integrals that are important in statistics.
Task 3
A person blows up a balloon. Assume that the balloon is spherical and that the person blows so that the volume increases at a rate of 2 L/s = 2000 cm3/s (2 liters per second).
a) Find an expression for the radius r of the balloon expressed in terms of the volume V .
b) Since the volume V increases, the radius r will also increase. Find an expression for the growth
rate to the radius. Hint: it is easier to derive see 'speed-coupled' tasks' in the book.
c) How fast does the radius increase when the volume is 4 L?
Task 4
Two people start running at the same time from the same place. One person runs west at a speed of 3 m/s and the other runs north at a speed of 4 m/s.
a) How far apart are the two people after 5 seconds?
b) How fast does the distance between the two people increase after 5 seconds?
Task 5
Consider the equation
(a) Find the general solution of the equation.
(b) Find the solution that satisfies the initial condition y(0) = 2 .