This is specifically for 20130321, but is likely to grow, simply because five questions does not a webpage make1.
DJS 20130320
FPM Examples 6
1. Differentiate y = e5x + 8e-4x wrt x;
Find and classify the stationary value (x and y).
2.Show that e3ln2 = 23.
Integrate y = e5x + 8e-4x wrt x between ln 2 and ln 4, giving an exact rational answer.
3. Solve 2 cos (5θ - π/6) = 1 for 0<θ<π
4.A circle has area 16 sq units.
a) What is the length of the circumference of the circle?
b) Develop the formula for the perimeter of a sector.
c) What angle is subtended by a sector of perimeter of 15 units? Exact answers will do nicely. Show that this is near to three quadrants.
5. a) Show that 4633 is a product of two prime factors.
b) Find exactly tan 2φ if cos φ = 36/85 and express it as a product of prime factors,
2a3b5c7d etc where a,b,c,d are integer.
6. Write down the 3 expansions of cos 2θ; you have been told repeatedly to learn these!
7. Find an expression for tan 3θ entirely in terms of tan θ.
8. A portion of a large hill can be modelled as a rectangular-based pyramid ABCDE. Base plane ABCD has E vertically above A. AB=18, BC=39, CE=2. Units are kilometres.
a) Find length EC
b) Find angle CEB to the nearest degree
c) A climb of tan-10.04 is the upper limit for some traffic. If a straight route at this limit starts at C, going up face CBE, and reaches CE first at point P [vertically above point Q on BC], what distance is BP? 2
1 “One swallow doth not a summer make” from Aristotle, presumably in translation: "One swallow does not a summer make, nor one fine day; similarly one day or brief time of happiness does not make a person entirely happy."
2 Call BQ x and PQ h. CQ=25h. Draw triangles APQ, CQB and BPQ. BPQ is similar to BEA.Find h, find BP
DJS 20130402