MATHS1001 - Modelling and Change Assignment 3 - Federation University

Assignment Task

Question 1

For the function f f (x) = x ³ − 8x ² − 35x  + 10 determine:

1. intervals where ff is increasing or decreasing,

2. local minima and maxima of ff,

3. intervals where ff is concave up and concave down, and

4. the inflection points of ff.

Question 2

An object is travelling along the graph of the function f f (x ) = x ³ (in the first quadrant), its x -coordinate is increasing at a constant rate of 15 cm/sec. Find how quickly the angle of elevation between the origin and the object is changing when x  = 5 cm.

Question 3

Find the horizontal asymptote(s):

f f (x ) = 4x ³ + 2x  − 1/ 3|x | ³ + 5x 

Question 4

Evaluate the following limits:

a) lim ex2/ 5 − x  ³

b) lim 1/sin x - 1/x

Question 5

An object is moving with the given velocity. Find the position function of the object.

v(t) = 1/3√t + cos(πt), s(4) = 3

Question 6

Monthly repayments M(r ) on a 360-month (30 year) $500,000 home loan are given by

Here r represents an annual interest rate (expressed as a decimal). Use the Newton’s method to calculate the interest rate that allows to make repayments of

$2500 a month. In addition to the final answer, show the detailed calculations for the first iteration. Round to 5 decimal places.

Question 7

Find the following derivative

d/dx 5 cos √x  (2t ³ + e t) dt

Differentiating the integral directly;

Evaluating the integral first and differentiating the result

Question 8

Solve the following integrals:

a) x − 3 / x2 − 6xdx

b) e^π/4 5/x(1 + ln2 x)

Question 9

A circle-shaped oil spill has a density of 50/(1 + r) kg/m ² at a distance of r kilometers from the center of the spill.

• Write a Riemann sum that approximates the total mass of the spill, if its radius is r = 2000m. (Assume that the thickness of the spill is negligible)
• Find the exact value of the total mass of the

Question 10

Lorenz curve y = f f (x) is used to represent a wealth distribution in a nation. The entire population is plotted along the x-axis from poorest to richest, and the height of the curve on the y-axis shows the percentage of total income received by the proportion of the population given on the x-axis. Graph of function y = x, called the line of perfect equality, represents the population where everyone has the same income. The area between the graphs of the two functions for 0 ≤ x ≤ 100 shows the country’s income inequality.

The table below gives the percentage of income for selected percentages of people in a country.

• Use Excel to fit a quadratic curve to the given An example and instructions are provided here.
• Plot the data along with the obtained curve above and the line y = x
• Using the obtained curve, approximate the income inequality for the given

Question 11

Set up and evaluate the integral that represents the volume of the solid of revolution formed by revolving around the axis y = −4 the region in the first quadrant bounded by y = 1/3 x + 2, y = 0 and x = 8.

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