Using the Gradient Function (Derivative)

Consider the function,
 $f\left(x\right)\ =\ 3x^2\ +\ 2x\ +\ 5$ 

Find the gradient function. 

•  $f'\left(x\right)\ =\ 6x\ +\ 2x\ +5$  

•  $f'\left(x\right)\ =\ 6x\ +2$  

•  $f'\left(x\right)\ =\ 6x^2\ +\ 2x\$  

•  $f'\left(x\right)\ =\ 6x\ +\ 7$  

Consider the function,
 $f\left(x\right)\ =\ x^2\ -13x$ 

Find the gradient when x = 0.

•  $f'\left(0\right)\ =\ -13$  

•  $f'\left(x\right)\ =\ 2x\ -\ 13$  

•  $f'\left(0\right)\ =\ -11$  

•  $f'\left(0\right)\ =\ 0$  

Consider $y=\frac{5}{x^2}$  
Find the gradient when x = -1

•  $f'\left(-1\right)\ =\ -10$  

•  $f'\left(-1\right)\ =\ 5$  

•  $f'\left(x\right)\ =\frac{-10}{x^3}$  

•  $f'\left(-1\right)\ =\ 10$  

Consider the function,
 $f\left(x\right)\ =7-8x^2\ +\ 2x^3$ 

Find the gradient function. 

•  $f'\left(x\right)\ =\ 7-16x\ +6x^2$  

•  $f'\left(x\right)\ =\ -16x\ +6x^2$  

•  $f'\left(x\right)\ =\ -16x^{2\ }+\ 2x^3$  

•  $f'\left(x\right)\ =\ -16x^2\ +2x^2$  

Consider the total cost,
 $C\left(q\right)\ =\ 8q^2-9$  
Determine the marginal cost  when q = 10.

•  $C'\left(10\right)\ =\ 160$  

•  $C'\left(10\right)\ =\ 151$  

•  $C'\left(10\right)\ =\ 791$  

•  $C'\left(10\right)\ =\ 1600$  

Consider the total cost,
 $C\left(q\right)\ =\ 300\ +\ 5q^2$  
Determine the marginal cost  when q =120.

•  $C'\left(120\right)\ =\ 900$  

•  $C'\left(120\right)\ =\ 1500$  

•  $C'\left(120\right)\ =\ 72,300$  

•  $C'\left(120\right)\ =\ 1200$  

A function is defined as, $f\left(x\right)\ =\ x^2-4x\ -\ 12$  
Find the value of x on the curve where the gradient is 8.

• x = 6

• x = 2

• x = 0

• x = 2.5