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b) Use the formula for the area of a sector A = \frac{1}{2}r^2 \theta c) Use the formula for length of an arc l = r\theta Remember that \theta must be in radians for those two formulae. d) Use the alternate formula for area of a triangle A = \frac{1}{2} ab \sin C (or alternatively...

Okay, let me elaborate then. We can rewrite the relation to make it a bit less confusing. |y| = f(x) Using one of the definitions of the absolute value function |k| = \sqrt {k^2} , \sqrt{y^2} = f(x) y^2 = f(x)^2, \, f(x) > 0 (when we square both sides, we unintentionally define the...

To prove that two lines intersect at a point, the only thing you need to do is show that the point satisfies the equations of both of the lines. There's no need to solve the equations if the answer is already given to you.

V = \pi \int^1_0 (3-3^x)^2 \, dx = \pi \int^1_0 9 - 6(3^x) + 3^{2x}\, dx = \pi \int^1_0 9 - 6e^{x \log 3} + e^{2x \log 3}\, dx = \pi \left[ 9x - \frac{6}{\log 3}e^{x \log 3} + \frac{1}{2 \log 3}e^{2x \log 3} \right]^1_0 = \pi \left[\left(9 - \frac{6}{\log 3}e^{\log 3}+ \frac{1}{2...

Imagine the graph of any function y = f(x) on the Cartesian plane, for example f(x) = x^2 . The graph of a function is a visualisation of the relationship between an input x and an output y . 1) y = -f(x) This can be rewritten as -y = f(x) . You're replacing y with -y , swapping...

The period of \sin(x) is 2\pi. So, \sin\left(\frac{11\pi}{4}\right) = \sin\left(\frac{11\pi}{4} - 2\pi \right) = \sin\left(\frac{3\pi}{4}\right) Also, \sin(x) = \sin(\pi-x) . So, \sin\left(\frac{3\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt 2}{2} If you want the answer...

The sine wave has symmetry across the origin, so it's an odd function, which means that f(-x) = f(-x). Basically that implies that the sine of a negative angle is the same as reversing the sign of the sine of the positive angle, i.e. \sin -\theta = -\sin \theta . When we go in the...

Re: Can someone please do the first one log2 x showing working and I'll do the other y = \frac{1}{\log 2} \left(\frac{x}{3} - 1\right) + \log_2 3 y = \frac{1}{\log 2} \left(\frac{x}{3} - 1\right) + \frac{\log 3}{\log 2} y = \frac{1}{\log 2} \left(\frac{x}{3} - 1 +\log 3\right)

Infinity. If bounds are given, use the formula V = \pi \int^a_b y^2\, dx and use k = e^{\log k} (for all k > 0) (remember to give your answers in cubic units!)

The question is somewhat ambiguously worded but I will assume the meaning to be "decreasing by 5% of the previous year's population". Use the formula for geometric sequences. Let T_n be the population, expressed as a percentage of the original, at the start of the n th year (be careful...

Re: Please help with these three questions they the only three I couldn't do from the Evaluate \log e (keeping in mind that logarithms are implicitly in base e).

7a) The integral \int^2_0 e^{-x} dx gives the area UNDER the curve. You need to subtract that from 2 (area of rectangle) to get the correct answer. 2 - \int^2_0 e^{-x} dx \\ \\= 2 - [-e^{-x}]^2_0 \\\\ = 2 - (-e^{-2} - (- 1)) \\\\ = 1 + e^{-2} 22b) A = \pi \int^2_0 (\sqrt{2x}...

Q9) Construct lines: Let AP = PQ = QR = x (or any pronumeral of your choice) Remember that \tan \angle BAR = \frac{BR}{AB} and that AB, BR can both be written as sums of intervals. ____________________________________________________________ Q10) This question becomes trickier if...

8a) Because P(x) is monic and of degree 4, the leading term is x^4. P(x) is also even, so no terms of odd degree can be present. Therefore our expression is P(x) = x^4 + cx^2 + d. 8b) We know that x = 3, P(x) = 0, and x = 1, P(x) = 8 . From this, you can substitute into the general...

I don't like those big 7-markers where you must memorise and regurgitate a wall of text. It really kills off my enjoyment of Chemistry.

I've heard of something similar where you graph the numerator and denominator separately to make a "sign diagram" so you can tell which side of the y-axis the line will be. But that doesn't have anything to do with critical points.

It doesn't matter if your graph's scale isn't perfect, you don't need to measure out centimetres on your ruler. As long as you sketch a reasonable graph, showing any necessary features (including any that were asked for) e.g. intercepts, maxima/minima, pts. of inflexion you'll get the marks. The...

The perimeter of the rectangle is obviously 20cm. Let the sides be x and y , and then find y in terms of x . Once both sides are expressed in terms of x , you can obtain a formula for area which you can then differentiate. For reference, your area formula should be A = 10x-x^2 (oops!).

Very large/heavy atoms tend to be unstable. The radioactive elements are only found near the end of the periodic table, where the largest atoms reside - that's no coincidence. The nucleus is too big to keep itself together and will lose particles through radioactive decay. AFAIK a net positive...

(i) P(3 red) = 25/75 * 24/74 * 23/73 Clearly, the ticket that wins the first draw has a 1/3 chance of being red. When that ticket is removed there are 24 red tickets, 25 blue and 25 green, making 74 tickets in total. Therefore the next ticket drawn has a 24/74 chance of being red. After the...