Chapter 6: Sections 6.1, 6.2.1, Chapter 8: Section 8.1, 8.2 and 8.5. In Business world the study of change important

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1 Study Unit 5 : Calculus Chapter 6: Sections 6., 6.., 6.. Chapter 8: Section 8., 8. and 8.5 In Business world the study of change important Eample: change in the sales of a company; change in the value of the rand; change in the value of shares; change in the interest rate etc. Equally important is the rate at which these changes take place. Eample: If the sales of a company increased by R ,00, it is important to know whether this change occurred over one year, two years or ten years. Rate of change: changes over time, change in costs for different production quantities in a production process, etc. Change consists of two components: size and direction. Let s look at linear function : y = m + c. Slope (m) is the change in y which corresponds to change change in y y y of one unit in the value of : =. change in

2 Slope => indication of rate of change = constant Value of the slope => size of the change Sign of the slope => direction of the change. positive sign an increase; negative sign a decrease. Let s look at a non-linear function, for eample a quadratic function The size and direction are not constant, but change continuously. Use the mathematical technique of Differentiation to determine the rate of change.

3 . Differentiation Slope of a curve = change in y / change in = rate curve change Use differentiation to get slope at a given point dy/d = f () derivative of y with respect to Pronounce this as dee-y-dee-. Use rules of differentiation to obtain the derivative Many rules of differentiation look at just namely Power Rule: If f() = n, then f '() or d ( n ) n = n for n 0 d For eample n integer and positive d ( ) 4 4 d = = 4 4 n integer and negative d 4 ( ) = = = 4 d

4 4 n is a fraction and positive d = ( ) = = = d n is a fraction and negative d = ( ) = = = = d ( ) Note:. The derivative of any constant term say a, that is a term which consists of a number only, is zero: da 0 d =, where a is a constant. Eample: f() = 4 then f '() = 0.

5 5 d a f = d a f '().. ( ) Eample: f() = 7 5 then f '() = = If f( ) = g ( ) + h ( ), then f '( ) = g'( ) + h'( ). Eample: f() = then f '() = Eample: f() = then f '() = Steps:. First we need to simplify the given epression so that we can use the basic rule of differentiation.. Secondly we differentiate the new epression using the d n n basic rule = n wheren 0 of differentiation d For eample:.. d d ( ) ( ) d d 4 = = = = 4 d d = = ( ) = = = d d Discussion class eample 9

6 6 Question 9 Differentiate the following epression Solution First we need to simplify the given epression so that we can use the basic rule of differentiation ( 4 + 4) = = ( )( ) = ( ) = Net we can differentiate the new epression using the basic rule d d = where 0. Therefore n n n n + d d d 4 4 d ( = ) = =

7 7 Application : Minimum or maimum, verte, turning point => slope = 0 => dy/d = 0 Discussion class eample 4 (c) using differentiation Also called rate of change because slope is rate of change Slope of a tangent line at a given point derivative at that point. Marginal analysis Marginal revenue : MR = dtr/dq Marginal cost : MC =dtc/dq Discussion class eample 0

8 8 Question 0 What is the marginal cost when Q =0 if the total cost is given by: TC = 0Q 4 0Q + 00Q + 00? Solution The marginal cost function is the differentiated total cost function. Thus by differentiating the total cost function we can determine the marginal cost function. Now if the total cost function is TC = 0Q 4 0Q + 00Q + 00 then the marginal cost function is dtc MC Q Q dq = = Now the marginal cost function s value when Q is equal 0 is MC = 80(0) 60(0) + 00 = =

9 9. Integration Is the reverse of differentiation y d(y)/d d(y) d( y) o Indefinite integral : different rules Steps: Simplify function before you integrate write it so that you can apply the integration rule for eample o ( a + b) = a + b o = Apply basic integration rule n+ n d= + c where n n + 0+ a a d = + c = a + c where a is a constant Discussion class eample and Hint : test your answer: differentiate answer, must be equal to function integrated.

10 0 Question Evaluate the following ( + + )d Solution To integrate the function we make use of the basic rule of n+ n integration namely = + cwhen n. Therefore: n + ( + + )d = d + d + d = c = c = c

11 Question Determine Solution Q + d Q Q First simplify the function to be integrated: Q+ ( Q+ ) = Q Q = ( Q+ ) Q = Q + Q Integrate the function using rule n+ n = + cwhen n n + ( Q + Q ) dq= Q + Q + + Q Q = + + c + + Q Q = + + = + + Q Q c Q = + Q + c c

12 Definite integral : area under a given curve between two points a and b: a b n+ n+ n d = ( = b) ( = a) n+ n+ Steps :. Simplify the function. Integrate the function by applying the basic rule of integration. Calculate the value of the integrated function at the value a substitute the values a into the integrated function answer 4. Calculate the value of the integrated function at the value b substitute the values b into the integrated function answer 5. Subtract answer from answer Discussion class eample

13 Question Evaluate ( z+ ) dz Solution Integrate the function using the basic rule n+ n = + c n + n and substitute the values into the integrated function: ( z+ )dz = z + z + when z = ( + z) ( ) = ( + ) + ( ) = ( ) = ( ) = + =