Given that f(x)=cosx, show that (f(x+h)-f(x))/h=cosx((cosh-1)/h)+sinx(sinh/h)

usagirl007A 2021-05-28 Answered
Given that \(\displaystyle{f{{\left({x}\right)}}}={\cos{{x}}}\), show that \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={\cos{{x}}}{\left(\frac{{{\text{cosh}{-}}{1}}}{{h}}\right)}+{\sin{{x}}}{\left(\frac{{\text{sinh}}}{{h}}\right)}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

opsadnojD
Answered 2021-05-29 Author has 9358 answers
Substitute the given function: \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\frac{{{\cos{{\left({x}+{h}\right)}}}-{\cos{{x}}}}}{{h}}\)
Use the the sine of a sum identity: \(\displaystyle{\sin{{\left({A}+{B}\right)}}}={\sin{{A}}}{\cos{{B}}}+{\cos{{A}}}{\sin{{B}}}\) \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\frac{{{\cos{{x}}}{\text{cosh}{-}}{\sin{{x}}}{\text{sinh}{-}}{\cos{{x}}}}}{{h}}\)
Regroup as: \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}=\frac{{{\cos{{x}}}{\text{cosh}{-}}{\cos{{x}}}-{\sin{{x}}}{\text{sinh}}}}{{h}}\)
Separate as: \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={\left(\frac{{{\cos{{x}}}{\text{cosh}{-}}{\cos{{x}}}}}{{h}}\right)}-{\left(\frac{{{\sin{{x}}}{\text{sinh}}}}{{h}}\right)}\)
Factor out sinxsinx from the first term and cosxcosx from the second term:
\(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={\cos{{x}}}{\left(\frac{{{\text{cosh}{-}}{1}}}{{h}}\right)}-{\sin{{x}}}{\left(\frac{{\text{sinh}}}{{h}}\right)}\)
Have a similar question?
Ask An Expert
34
 
content_user
Answered 2021-08-11 Author has 1944 answers

It will help you

16

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-05-02
Given that \(\displaystyle{t}{\left({x}\right)}={\sin{{x}}}\), show that \(\displaystyle{f{{\left({x}+{h}\right)}}}-\frac{{f{{\left({x}\right)}}}}{{h}}={\sin{{x}}}{\left(\frac{{{\text{cosh}{-}}{1}}}{{h}}\right)}+{\cos{{x}}}{\left(\frac{{\text{sinh}}}{{h}}\right)}\)
asked 2021-06-22
\(\displaystyle\frac{{{1}+{\cos{{x}}}}}{{\sin{{x}}}}={\csc{{x}}}+{\cot{{x}}}\)
asked 2021-10-28
Show that the equation has exactly one real root.
\(\displaystyle{2}{x}+{\cos{{x}}}={0}\)
asked 2021-10-18

Find the solution of the differential equation that satisfies the given initial condition. \(dy/dx=x/y, y(0)=-3\)

asked 2021-10-14
Suppose \(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}{\quad\text{and}\quad}{g{{\left({x}\right)}}}=\sqrt{{{x}-{5}}}\), to find:
1. (fog)(x)
2. Domain of (fog)(x)
3. (gof)(x)
4. Domain of (gof)(x)
asked 2021-10-26
if x=2 , f(x)=1
if x=3 , f(x)=4
if x=5 , f(x)=-2
if x=8 , f(x)=3
if x=13 , f(x)=6
f is twice variable for all real numbers.
1. Find f'(4)
2. Approximate \(\displaystyle{\underset{{{2}}}{{\int}}}^{{13}}{f}'{\left({x}\right)}{\left.{d}{x}\right.}\)
2. Find the value of \(\displaystyle{\underset{{{2}}}{{\int}}}^{{8}}{\left({3}-{f}'{\left({x}\right)}\right)}{\left.{d}{x}\right.}\)
asked 2021-09-11
evaluate the function for the indicated values. \(\displaystyle{f{{\left({x}\right)}}}={\left[{x}\right]}.{f{{\left({2.1}\right)}}}\)

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...