for f(x)=6x³, construct and simplify the difference quotient (f(x + h)-f(x))/h. (f(x + h)-f(x))/h = □ (type…

for f(x)=6x³, construct and simplify the difference quotient (f(x + h)-f(x))/h. (f(x + h)-f(x))/h = □ (type in descending powers of x)

for f(x)=6x³, construct and simplify the difference quotient (f(x + h)-f(x))/h. (f(x + h)-f(x))/h = □ (type in descending powers of x)

Answer

Explanation:

Step1: Find f(x + h)

Given (f(x)=6x^{3}), then (f(x + h)=6(x + h)^{3}). Using the formula ((a + b)^{3}=a^{3}+3a^{2}b + 3ab^{2}+b^{3}), we have (f(x + h)=6(x^{3}+3x^{2}h + 3xh^{2}+h^{3})=6x^{3}+18x^{2}h + 18xh^{2}+6h^{3}).

Step2: Calculate f(x + h)-f(x)

(f(x + h)-f(x)=(6x^{3}+18x^{2}h + 18xh^{2}+6h^{3})-6x^{3}=18x^{2}h + 18xh^{2}+6h^{3}).

Step3: Simplify the difference - quotient

(\frac{f(x + h)-f(x)}{h}=\frac{18x^{2}h + 18xh^{2}+6h^{3}}{h}). Factor out (h) from the numerator: (\frac{h(18x^{2}+18xh + 6h^{2})}{h}). Cancel out the (h) terms, we get (18x^{2}+18xh + 6h^{2}).

Answer:

(18x^{2}+18xh + 6h^{2})