Year 12 Mathematical Studies Calculus Applications Test1. A particle P is undergoing straight line motion with distance from an arbitrary origin given by metres, where t is measured in seconds, a) Find expressions for the velocity and acceleration and draw sign diagrams of each.

b) Find the initial position of the particle.

c) What are the values of t for which the speed of the particle is decreasing?

d) Determine the time and position where P changes direction

e) Evaluate the distance covered by P in the first 3 seconds of motion.

2. The length of the leaf of a plant (in mm), t days after it emerges is given by: where a) Find the length of the leaf afteri) 3 days ii) 1 week

b) Evaluate algebraically how long it will take the leaf to grow to 12mm

c) Determine an expression for L(t) and hence evaluate the rate at which the leaf is growing after 3 days

d) Discuss L as t

e) Sketch the graph of L against t showing the details calculated in (a), (b) and (d) above.

3. A manufacturer has a factory which is capable of producing up to 60 bedroom suites per day. The total cost of materials and labour needed to make the suites is $, where . In addition there are fixed daily costs of $1200.

If x suites are made per day and each suite is sold for $:a) Show that profits, P(x) are given by the expression

b) Evaluate P(x) and hence determine x such that P(x) = 0

c) How many suites should be made and sold each day to maximise profits?

d) What will be the maximum profits in this case?

e) If production is cut back to a maximum of 40 bedroom suites per day, explain how many suites will maximise profits and what the maximum profit will be in this case.

4. Point P moves on the graph of y = f(x), where for x > 0. The graph below shows P in three different positions:

O is the origin and point Q is on the x-axis. Triangle OPQ has a right angle at Q.Let OQ be x units long and OP be L units long.a) (i) Show that

(ii) Find the position of P at which is minimised.

(iii) Show that when is minimised, OP is normal to f(x)

b) Suppose that point P moves on the graph of y = g(x), where for x > 0 and where b is a positive integer. Given that O is the origin, let OP be L units long.Show that, for all b, is a minimum when .