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Description | Marks out of | Weight (%) |
Assignment 5 | 20 | 10 |
Important: You must show all your working to receive full marks. Please note that all angles are given in degrees.
- Two buildings, one 12m high and the other 24m high, are directly opposite each other across a river. The angle of depression to the top of the smaller building from the top of the larger one is 52◦.
- Draw a diagram of the scenario.
- Calculate the width of the river. (Note that the angle of depression is the angle between the horizontal and your line of sight when looking down.)
(2 marks)
2. A sinusoidal function can be written in the form y = A sin (Bx), where |A| is the amplitude, and 2π/B is the period. For each of the following, state the function’s period and amplitude. |
a)y = − 1/5 sin 4x
b) y = 9 sin 3x/4
c) y = 2 cos 5x/6
(3 marks)
3. Find the six angles where 0 ≤ x < 2π such that cos (3x) = √1/2 .
4. For the function f (x) = x2 − 2x:
a) What is the derivative of f ?
b) Find f (x + h).
c) Express g(x) = (f (x + h) − f (x))/h, in terms of x and h.
d) Show that
f′(x) = lim g(x).
h→0
(4 marks)
4. Differentiate the following functions with respect to x:
(a) f (x) = x4 − 2x3 − x + 2
b) g(x) = ex − π2
c)w(x) = x−2 + 2 ln x
d) t(x) = 2 cos x − sin x
(5 marks)
6. For the function f (x) = x2 − 2x:
a) Find the equation for its tangent at x = 0.
b) Find the equation for its tangent at x = 1.
c) Find the equation for its tangent at x = 2.
d) What is special about f when x is one?
(4 marks)
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