Simplify the difference quotients between f(x+h)-f(x)/h and f(x)-f(a)/(x-a), if f(x)=sqrt(x^2-7)

Chardonnay Felix 2021-08-18 Answered
Simplify the difference quotients between \(\displaystyle{f{{\left({x}+{h}\right)}}}-\frac{{f{{\left({x}\right)}}}}{{h}}{\quad\text{and}\quad}{f{{\left({x}\right)}}}-\frac{{f{{\left({a}\right)}}}}{{{x}-{a}}}\), if \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}^{{2}}-{7}}}\)

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Expert Answer

estenutC
Answered 2021-08-19 Author has 11584 answers
\(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}^{{2}}-{7}}}\)
\(\displaystyle{f{{\left({x}+{h}\right)}}}=\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}\)
\(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\frac{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}-\sqrt{{{x}^{{2}}-{7}}}}}{{h}}\)
\(\displaystyle=\frac{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}-\sqrt{{{x}^{{2}}-{7}}}}}{{h}}\times\frac{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}-\sqrt{{{x}^{{2}}-{7}}}}}{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}-\sqrt{{{x}^{{2}}-{7}}}}}\)
\(\displaystyle=\frac{{{\left[{\left({x}+{h}\right)}^{{2}}-{7}\right]}-{\left[{x}^{{2}}-{7}\right]}}}{{{h}{\left[\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}+\sqrt{{{x}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{\left({x}+{h}\right)}^{{2}}-{x}^{{2}}}}{{{h}{\left[\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}+\sqrt{{{x}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{h}^{{2}}+{2}{x}{h}}}{{{h}{\left[\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}+\sqrt{{{x}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{h}+{2}{x}}}{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}+\sqrt{{{x}^{{2}}-{7}}}}}\)
\(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}^{{2}}-{7}}},{f{{\left({a}\right)}}}=\sqrt{{{a}^{{2}}-{7}}}\)
\(\displaystyle\frac{{{f{{\left({x}\right)}}}-{f{{\left({a}\right)}}}}}{{{x}-{a}}}=\frac{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}{{{x}-{a}}}\)
\(\displaystyle=\frac{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}{{{x}-{a}}}\times\frac{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}\)
\(\displaystyle=\frac{{{\left({x}^{{2}}-{7}\right)}-{\left({a}^{{2}}-{7}\right)}}}{{{\left({x}-{a}\right)}{\left[\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{x}^{{2}}-{a}^{{2}}}}{{{\left({x}-{a}\right)}{\left[\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{\left({x}-{a}\right)}{\left({x}+{a}\right)}}}{{{\left({x}-{a}\right)}{\left[\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}\right]}}}\)
\(\displaystyle=\frac{{{x}+{a}}}{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}\)
\(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\frac{{{h}+{2}{x}}}{{\sqrt{{{\left({x}+{h}\right)}^{{2}}-{7}}}+\sqrt{{{x}^{{2}}-{7}}}}}\)
\(\displaystyle\frac{{{f{{\left({x}\right)}}}-{f{{\left({a}\right)}}}}}{{{x}-{a}}}=\frac{{{x}+{a}}}{{\sqrt{{{x}^{{2}}-{7}}}-\sqrt{{{a}^{{2}}-{7}}}}}\)
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