Question #065bf

Question

202 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-19T05:59:48

# 21x^2+21xh+7h^2+3.#

Let us denote, by #Q_f,# the Quotient Ratio for a function (fun.) #f.#

Then, #Q_f" is given by, "Q_f={f(x+h)-f(x)}/h, h ne 0....(ast).#

We need #Q_f# for the fun. #f(x)=7x^3+3x.#

# f(x)=7x^3+3x rArr f(x+h)=7(x+h)^3+3(x+h).#

Now, #(a+b)^3=a^3+b^3+3ab(a+b).#

#:. f(x+h)=7{x^3+h^3+3xh(x+h)}+3(x+h),#

#=7{x^3+h^3+3x^2h+3xh^2}+3(x+h),#

#=7x^3+21x^2h+21xh^2+7h^3+3x+3h.#

# rArr f(x+h)-f(x),#

#=7x^3+21x^2h+21xh^2+7h^3+3x+3h-(7x^3+3x),#

#=21x^2h+21xh^2+7h^3+3h,#

#=h(21x^2+21xh+7h^2+3).#

#:.," by "(ast)," the Reqd. "Q_f=21x^2+21xh+7h^2+3.#