chapter 9: Application of Differentiation

Chapter 9: Application of Differentiation

9.1 Tangent and Normal

9.2 Extremum Problem

9.3 Application of Differentiation in Economics and business

9.1 Tangent and Normal Equation

•  $M_T=f'\left(x\right)=\frac{dy}{dx}$  

•  $M_N=-\frac{1}{M_T}$  

• Equation  $y-y_1=m\left(x-x_1\right)$  

9.2 Extremum problems

•  $f'\left(x\right)=\frac{dy}{dx}=0$  to find stationary point(s)

• 2 method to find the nature of stationary points ( first derivative and second derivative)

Method 1 (First Derivative)

Method 2 (2nd Derivative/ f''(x))

•  $f''\left(x\right)>0\$  minimum point

•  $f''\left(x\right)<0$  maximum point

•  $f''\left(x\right)=0$  (test failed ) back to first derivative test)

Method 2 (second derivative)

9.3 Application of Differentiation in Economics and business

•  $\overline{C\left(x\right)}=\frac{C\left(x\right)}{x}$  (Average Cost) 

•  $R\left(x\right)=p\left(x\right).x$  

•  $\pi\left(x\right)=R\left(x\right)-C\left(x\right)$  

Multiple Choice

Write the equation of the normal line of:  $f\left(x\right)=2x^2-\frac{1}{x}$  at x = 1

•  $y=-\frac{1}{5}x+\frac{4}{5}$  

•  $y=5x-6$  

•  $y-1=5\left(x-1\right)$  

•  $y-1=-\frac{1}{5}\left(x-1\right)$  

Multiple Choice

What is the equation of the line tangent to f(x)= 4x²+2x-1 at x=0?

• y+1=2(x+1)

• y=2x-1

• y=2x+2

• y= -x-1

Multiple Choice

Local maximum point is obtained when

• f ' (x) exchange from negative to positive

• f '' (x) exchange from positive to negative

• f ' (x) exchange from positive to negative

• f '' (x) exchange from negative to positive

Multiple Choice

Which of the following is the equation for figuring out PROFIT?

• Profit = Revenue - Cost
• Profit = Cost - Revenue
• Revenue = Profit - Cost
• Costs = Profit - Revenue

Multiple Choice

Which of the following is the best definition of PROFIT?

• The total amount of income a business makes from selling products or services
• The amount of money a business has left over after paying for their costs.
• The total amount of money a business spends.

Multiple Choice

What is the stationary points for the curve f(x) = x³ +3x² - 24x and determine their nature.

• Maximum point : (2, -28) and minimum point : (-4, 80)

• Minimum point : (2, 28) and maximum point : (-4, 80)

• Minimum point : (2, -28) and maximum point : (4, 80)

• Minimum point : (2, -28) and maximum point : (-4, 80)

Multiple Choice

What is the stationary point for the curve y=x² - 4?

• A minimum at ( 0, -4)

• A maximum at (0, -4)

• A minimum at (0,4)

• A maximum at (0,4)

Multiple Choice

 If (a,b) is a local minimum, then what will be true about f'(a)?

• It's positive
• It's negative
• It's zero
• Cannot be determined

Multiple Choice

The revenue function for selling x kg of butter cake is given by

 $R\left(x\right)=30x-x^2.$  the total cost to produce xkg of butter cake is given by the cost function  $C\left(x\right)=10x+20$  . Find the maximum profit

• RM 100

• RM 90

• RM80

• RM 120