Answer.AI
Question
Mathematics
If f ( x ) = 4 x 2 f(x)=-4x^(2), simplify: f ( 2 + h ) f ( 2 ) h (f(-2+h)-f(-2))/(h). Show all steps neatly.
See Answers
Get the Answer.AI App
Solve problem with AI
Best Answer
We are given the function f ( x ) = 4 x 2 f(x)=-4x^(2) and we need to simplify the expression f ( 2 + h ) f ( 2 ) h (f(-2+h)-f(-2))/(h).
Substituting x = 2 x=-2 into the equation for f ( x ) f(x), we get:
f ( 2 ) = 4 ( 2 ) 2 = 16 f(-2)=-4(-2)^(2)=-16
Substituting x = 2 + h x=-2+h into the equation for f ( x ) f(x), we get:
f ( 2 + h ) = 4 ( 2 + h ) 2 = 4 ( h 2 4 h + 4 ) = 4 h 2 + 16 h 16 f(-2+h)=-4(-2+h)^(2)=-4(h^(2)-4h+4)=-4h^(2)+16 h-16
Now, we can substitute these values into the expression we need to simplify:
f ( 2 + h ) f ( 2 ) h = ( 4 h 2 + 16 h 16 ) ( 16 ) h (f(-2+h)-f(-2))/(h)=((-4h^(2)+16 h-16)-(-16))/(h)
Simplifying the numerator, we get:
4 h 2 + 16 h 16 + 16 h = 4 h 2 + 16 h h (-4h^(2)+16 h-16+16)/(h)=(-4h^(2)+16 h)/(h)
Factoring out 4 h 4h from the numerator, we get:
4 h ( h + 4 ) h (4h(-h+4))/(h)
Canceling the factor of h h in the numerator and denominator, we get:
4 ( h + 4 ) = 4 h + 16 4(-h+4)=-4h+16
Therefore, the simplified expression is 4 h + 16 -4h+16.
You might be interested in...
Explore more...
Get the Answer.AI App
Solve problem with AI