MATH 130 -- Assignment 1

1999 D1 & E1, Mathematics IE

Due:

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Calculus

Question 1        (Answer)

Consider the curve y = x².

(i)

Write down $${\frac{\dD y}{\dD x}}$$.

(ii)

Evaluate $${\frac{\dD y}{\dD x}}$$ at the points A(-1, 1) and B(2, 4).

(iii)

Find the equations of the tangents at the above two points.

(iv)

Find the point P of intersection of the above two tangents.

(v)

Show that the distance from P to A is $${\frac{3\sqrt 5}{2}}$$.

Question 2        (Answer)

A train moves a long a track in the shape of a parabola y = x² (distances in kilometres). A road runs along the x-axis and there is a station at the point where the track and the road come together.
You are in a house 1 km from the station along this road, and as the train approaches the station, the light on the front of the train shines directly in the front window.
How many metres away is the train when this occurs (to the nearest 100 metres)?
(HINT: Let the train be at a point (a, a²). Work out the equation of the tangent at this point.)

$$\hbox{%% \ifx\pdfoutput\relax \WARMprocessMMA{ass1q2}{eps}{m... ...es {road} ,(-.5,1)*+!D\txt\large\bfseries {$y=x^2$} \end{xy}}$$

Algebra

Question 1        (Answer)

Simplify the following expressions:

(a)

15 + (- 4)(- 3) - 11 ÷ (- 3)^-1 x 5 - (- 2)

(b)

3x - {2y - 3(2x + 5y) - [7x - (2y - 3x)]}

(c)

(2a)³ ÷ (a/5)^-3

(d)

$${\frac{\displaystyle\frac1a+\frac1b}{\displaystyle\frac1a-\frac1b}}$$

Question 2        (Answer)

Expand (2x² - 3x + 5)(3x - 2) - (5x - 1)(x + 2)(x - 1) + 2(8x² - 17x + 6), then simplify it and factor it into the product of three linear terms.

Question 3        (Answer)

The area of a rectangle is 240 cm and its perimeter is 64 cm. Express these two facts algebraically in terms of the dimensions l and b of the rectangle.
Use the second fact to find an expression for l in terms of b, and hence find the dimensions of the rectangle.

Question 4        (Answer)

Pick any number. Square it.
Add on three times your original number. Add another two.
Divide the answer by two more than your original number.
Subtract the number you first thought of.
The answer is 1.

Explain why this is so.

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