Question

Given that f(x)=6x²−3x and c=1, find f′(c) by forming the difference quotient, (f(c+h)−f(c))/h, and taking the limit as h→0

a) 12
b) 3
c) 9
d) 0
e) − 3

Answer 1

To find f'(c), we need to form the difference quotient (f(c+h)−f(c))/h and take the limit as h approaches 0. By substituting the given values into the difference quotient and simplifying, we find that f'(c) = 9.

Step-by-step explanation:

To find f'(c), we start by forming the difference quotient:

(f(c+h)−f(c))/h

Given that f(x) = 6x²−3x and c = 1, we substitute these values into the difference quotient:

(f(1+h)−f(1))/h

Simplifying the expression, we get:

(6(1+h)²−3(1+h)−(6(1)²−3(1)))/h

Expanding and simplifying further:

(6(1+2h+h²)−3−3h−(6−3))/h

Combining like terms:

(6+12h+6h²−3−3h−6)/h

Simplifying the numerators:

(6h²+9h)/h

Cancelling out h in the numerator and denominator, we get:

6h+9

Now, we take the limit as h approaches 0:

lim(h→0) 6h+9 = 9

Therefore, f'(c) = 9.