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FINAL EXAM

MTH1010 Functions and their Applications 2016

1 SECTION A (80 MARKS)

1. Does the graph of the relation in the diagram represent a function? If not, then state which part of the function deÖnition is being violated.

2. Find all values of x such that < 2 .

3. What is the domain and range of the function g(x) =

4. The function f (x) = 2xx__3(1) has an inverse f__1 (x) = cx(ax)d(b): Find the four numbers a;b;c and d: Show all your working and specify the domain and range off__1 .

5. Consider the function f (t) = |2t - 1| - 1: Carefully sketch the graph of the function (including all intercepts).

6. Completely factorise the cubic

g(x) = x3 - 5x2 - x + 5

7. Identify and characterise the asymptotes and/or holes of the function f(x) = :

8. Evaluate the limit (you do not need to demonstrate the limit laws in this question)

xl_i2

9. Using the addition formula cos(θ + φ) = cosθ cosφ - sinθ sinφ and the Pythagorean identity, prove the double angle formulaecos2θ = cos2 θ - sin2 θ and cos2θ = 2cos2 θ - 1

10. Find

(a) lim t!1

(b) lim t!1+ jt

11. Using the exponential laws solve the following equation for all values of x such that

22+x ÷ 2x2 _4

12. For the function f(x) = sinx give the sequence of transformations (in a correct order) such that the Önal function obtained is

f(x) = - - 1

13. Identify the period, amplitude, midline and phase shift of the function

f(x) = - - 1

14. On thesetofaxes provided give a sketch of the function f(x) = - - 1.

15. For the graph indicated below identify an exponential function that Öts the curve

16. Let f(x) = : Find f/ (x) by first principles i.e. by Önding the limit f/ (x) = hl_i0

17. Directly evaluate ex2 _x 、and find the values for x for which ex2 _x is an increasing function.

18. At what point on the curve y = (x - 2)(x - 3) is the tangent to the curve perpendicular to the line 3x + y + 5 = 0? Give a sketch of the curve, the tangent and the perpendicular line on the axes provided below (make sure you show the point on the curve that gives rise to the tangent required).

19. Find the indefinite integral l (1 + sin(4x))dx:

Hence evaluate the definite integrall0 π/2 (1 + sin(4x))dx:

20. Find the value of a such that l0 1

3 - 2x

= ln,a for some a > 0:

2 SECTION B (80 marks)

1. You are given a choice between two phone plans. One plan is provided by company A and the other by company B.

Let CA be the monthly cost of using company A (in cents). The cost of data is $25:00 per month (base charge) and a charge of 2:5c per data transfer on top of the base charge.

Let CB be the monthly cost of using company B (in cents). The cost of data is $40:00 per month (base charge) and a charge of 2:0c per data transfer on top of the base charge.

If x is the symbol for the number of messages per month that are being sent then express the following:

(a) Write down the rule expressing CA as a function of x

(b) Write down the rule expressing CB as a function of x

(c) On thesetofaxes provided draw graphs for both functions CA and CB (Note: You will be required to determine IF CA = CB for some value of x)

(d) From these graphs determine which company you would choose if the total number of data transfers per month on your phone is

(i) 2000 transfers (ii) 3000 transfers (iii) 4000 transfers

2. Suppose the long term carrying capacity of a large termite mound is 35 million. In 2005 the termite population was about 5 million. By using the logistic growth model

P(t) = 1 + Ae__kt

where t is measured in years after 2005. Give the associated model information

(i) P(t) approaches 35 (million) ast gets large.

(ii) The population in 2005 was estimated at 5 (million).

(iii) The population in 2013 was estimated at 7 (million).

(a) Find the exact value of the constants M;A and k:

(b) What do each of these constants represent in the model?

(a) Use the following Ögure and Pythagorean identity to show that on the UNIT circle

(Hint: In the diagram draw a similar triangle reáected in the ① -axis)

(b) If

unit circle.

4sin[ (t - 8)] = -2sin

(d) The power requirements P1 and P2 of two towns are given by

P1 (t) = 3 - 2sin P2 (t) = 5 - 2,3

where t is measured in hours and 0 ≤ t ≤ 24:

By using the identity given in part (c) show that the total power consumption P(t) in both

towns is

PT (t) = P1 (t) + P2 (t) = 8 + 4sin[ (t - 8)]

(e) Plot the graph of PT (t) for 0 ≤ t ≤ 24 on thesetofaxes provided below.

(f) By referring to your graph in part (e) or by algebraic calculation Önd at which times

PT (t) achieves a maximum. What is the value of the maximum of PT (t)?

(g) By referring to your graph in part (e) or by algebraic calculation Önd at which times

PT (t) achieves a minimum. What is the value of the minimum of PT (t)?

4. Consider the polynomial f(①) = ①3 (4 - ①)

(a) Find the ① - and g- intercepts for the function f.

(b) By Önding the Örst derivative off / (①) identify its stationary points.

(c) Use the Örst derivative test to characterise these stationary points as local maxima or local minima (or neither a max nor min).

(d) Identify the intervals of the domain of f for which the function is increasing (f/ (①) 持 0)

and similarly Önd those intervals for which the function is decreasing (f/ (①) < 0).

(e) By Önding the second derivative f"(①) identify the points at which f"(①) = 0.

(f) Given that the necessary condition for the existence of a point of ináectionis that f"(①) = 0 for some value of ①) what is the su¢ cient condition for the existence of a point of

ináection?

(g) Locate the points of ináection for f(①).

(h) Using the information gathered from the previous parts sketch the graph of g = f(①) on

the set ofaxes provided below.