Subsidiary Mathematics

Key unit competence

Extend the concepts of functions to investigate fully logarithmic and exponential functions and use them to model and solve problems about interest rates, population growth or decay, magnitude of earthquake, etc.

Introductory activity

From the discussion, the function Ft found in c) and the function ( )YF found in d) are respectively exponential function and logarithmic functions that are needed to be developed to be used without problems. In this unit, we are going to study the behaviour and properties of such essential functions and their application in real life situation.

2. 1 Logarithmic functions

2.1.1 Domain of definition for logarithmic function

2.1.2 Limits and asymptotes of logarithmic functions

2.1.3 Continuity and asymptote of logarithmic functions

2.1.4. Differentiation of logarithmic functions

2.1.5 Variation of logarithmic function

Thus, 1110 1 1xxx−=⇔ =⇒=. If 1,x=11 ln1 1yf= =−=, thus (1,1) is a point of the graph.Variation table of lny fx x x== −From the table, one can observe that the function is decreasing for values when x lies in ]0,1] and increasing for x greater than 1. The point (1,1) is minimum or equivalently the function takes the minimum value equal for x=1. The minimum value that is equal to 1 is absolute.

2. 2 Exponential functions

2.2.1 Domain of definition of exponential function

2.2.2 Limits of exponential functions

2.2.3. Continuity and asymptotes of exponential function

2.2.4. Differentiation of exponential functions

2.2.5 Variations of exponential functions

2. 3 Applications of logarithmic and exponential functions

Logarithmic and exponential functions are very essential in pure sciences, social sciences and real life situations. They are used by bank officers to deal with interests on loans they provide to clients. Economists and demographists use such functions to estimate the number of population after a certain period and many researchers use them to model certain natural phenomena. We are going to develop some of these applications.

2.3.1 Interest rate problems

If a principal P is invested at an interest rate r for a period of t years, then the amount A (how much you make) of the investment can be calculated by the following generalised formula of the interest rate problems:

2.3.2 Mortgage problems

2.3.3 Population growth problems

2.3.4 Uninhibited decay and radioactive decay problems

2.3.5 Earthquake problems

2.3.6 Carbon dating problems

2.3.7 Problems about alcohol and risk of car accident

END UNIT ASSESSMENT