Reference no: EM132352406

Question 1 - (a) The exponential function that is used most frequently in mathematics is the function y = e^x, where e = lim_n→∞(1+(1/n))ⁿ. Using Excel, complete a table of values and graph this function for large values of n, Discuss the behavior of this graph as n increases towards ∞.

(b) Using technology, graph the three functions on the same axes: f(x) = 2^x and g(x) = 3^x and h(x) = e^x.

i. Find the point(s) of intersection and confirm algebraically.

ii. For what values of x is f(x) > g(x).

iii. Which graph lies between the others and explain why it does?

iv. What happens as x → ∞ or x → -∞?

v. What is the Domain and Range for each function?

Question 2 - (a) The functions f(x) = 10^x and f^-1(x) = log₁₀x are the inverse functions of each other. Complete the table of values below and explain how the values in the table illustrate this statement.

x	y = f(x) = 10x		x	y = log10x
-3			1/8
-2			1/4
-1			½
0			1
1			2
2			4
3			8

(b) Sketch the graphs of y = 10^x, y = log x and y = x in the same coordinate system and state the domain and range of the common logarithmic function.

(c) Sketch the graph of y = e^x, y = In x and y = x in the same coordinate system and state the domain and range of the natural logarithmic function.

Question 3 - The table below shows the decay of the isotope Nitrogen-13, commonly used in medical and biological experiments.

You have been assigned unique data for the activity section of the below table, given to you by your teacher.

Time (seconds)	Activity (Decay/second)
	Riley	Maybellina	Zindendria	Hannah	Larissa	Sydney
60	1150	460	698	1395	930	1630
120	1095	436	650	1300	870	1525
180	1015	405	612	1225	810	1420
240	950	380	565	1140	755	1325
300	870	345	532	1059	705	1236
360	823	332	495	992	660	1150
420	775	305	460	920	615	1075
480	718	287	437	865	575	1000
540	669	265	400	801	535	935

Develop a mathematical model that gives the activity (A) after time (t) seconds. Use this mathematical model to work out the Half-Life of Nitrogen-13, the time take for the activity to reduce by a factor of 1/2.

Question 4 - Use your knowledge of Question 4 to develop a second model for the total number of decays over time.

Question 5 - P(a, b) is a variable point in the first quadrant on the curve y = e^-x. Q and R are the points where the tangent to the curve at P meets the y-and x-axes respectively.

a) Use algebraic methods to find the maximum area of triangle OQR, where O is the origin.

b) If the point (a, b) is not confined to the first quadrant is there another possible answer?

Note - In question 3 and 4 you must use the data for Larissa.