MATH 130 -- Assignment 2

1999 D1 & E1, Mathematics IE

Due:

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Calculus

Question 1        (Answer)

Consider the curve $\mathcal {C}$, given by the equation y² = 48x² - x⁶ + 97.

(i)

Show that only one of the following points lies on $\mathcal {C}$:
(0, 9), (1, 12), (1, 6), (-1,-6), (10, k).

(ii)

Find $${\frac{\dD y}{\dD x}}$$ at that point.

(iii)

Find the points on $\mathcal {C}$ where the slope is 0.

Question 2        (Answer)

A large artificial waterfall is 100 metres long and has a cross-section in the shape of the cubic y = 6x² - x³ + 8 between x = -3 and x = 5, where the distances are in metres.
The ends of the pool are closed so that a pool can form as shown.

$$\hbox{%% \ifx\pdfunknown\relax \WARMprocessMMA{ass2q2}{eps}... ...l} \xyMarkedPos{height}+(0,40)*!L{y=6x^2 - x^3 + 8} \end{xy}}$$

(i)

Find the maximum depth of water that can form in the pool.

(ii)

Show that the breadth of the pool, when full, is 6 metres.

(iii)

Find the volume of the pool, in Megalitres, when it is full.
(1 cubic metre is 1000 litres, and 1 Megalitre is a million litres.)

Question 3        (Answer)

You are at a point Y in a desert. An East-West road runs 8 km to the north and the nearest point to you is R.
There is a motel on the road at M, 6 km east of R.
Suppose you can walk at a rate of 5 km per hour along the road, but only 2 km per hour across the sand.

Suppose you head in a straight line to the point P on the road at x km to the east of R, and then walk along the road to M.
Let s be the time taken to cross the desert to P and let r be the time to reach M from P, along the road. (Both times are to be in hours.)

(i)

Draw a diagram showing the desert, the road, and the points Y, R, P and M. Label it with the appropriate distances (in terms of x).

(ii)

Show that s = ${\frac{1}{2}}$$\sqrt{x^2+64}$ and r = ${\frac{1}{5}}$(6 - x).

(iii)

Find $${\frac{\dD s}{\dD x}}$$ in terms of x and s.
(HINT: square s and use implicit differentiation.)

(iv)

Let t be the total time (in hours) to reach M, via P.
Find t in terms of r and s and hence find $${\frac{\dD t}{\dD x}}$$ in terms of x and s.

(v)

What is the shortest time required to reach the motel (in hours and minutes)?

How does this compare with the time taken to cross the desert directly to R and to walk the 6 km along the road, and the time taken to reach M by walking in a straight line across the desert?

Algebra

Question 4        (Answer)

(a)

Solve the following exponential and logarithmic equations:

(i)

($\sqrt{8}$)^x = 32.

(ii)

log(x²) + log 2 = log(5x + 3)

(b)

Consider the following statements for values a, b, c, d and x for which both sides of the equation make algebraic sense.
If the statements are false, give an example to show this. Otherwise, carefully indicate why they are true.

(i)

$${\frac{a}{b}}$$ + $${\frac{c}{d}}$$ = $${\frac{a+c}{b+d}}$$.

(ii)

x² - 3x + 4 > 1.

(iii)

$\sqrt[3]{x \sqrt x}$ = $\sqrt{x}$.

Question 5        (Answer)

(a)

Draw the graphs of cos 2x and cos x on the same axes for -3$\pi$$\le$x$\le$3$\pi$.
Use your graph to determine the number of solutions of cos 2x = cos x in this range.

(b)

Use a trigonometric identity to write cos 2x in terms of cos x.
Hence solve cos 2x = cos x algebraically, and list the solutions in the range -3$\pi$$\le$x$\le$3$\pi$.
[Does your answer agree with your answer in the previous question, and are the solutions as expected from your graph?]

(c)

Find the exact value of cos(${\frac{7}{6}}$$\pi$) - tan(-${\frac{9}{4}}$$\pi$) + sin(${\frac{11}{3}}$$\pi$)

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