Linear Approximation: Find the linear approximation for f(x) = x sin(pi x^2) about x = 2

[rolexsyrup1212]

 

[stapel]

Find the linear approximation for [imath]f(x) = x \sin(\pi x^2)[/imath] about [imath]x = 2[/imath]. Use the approximation to estimate [imath]f(1.99)[/imath].

Please reply with a clear listing of your thoughts and efforts so far, so we can see what's going on.

Thank you!

 

[Steven G]

f'(a) ~ [f(a+h)-f(a)]/h.
Why am I using an approximation symbol?
Now solve that approximation for f(a+h).
Show that work and we will continue from there

 

[pka]

View attachment 36560

This is from James Stewart's calculus textbook. [imath]\bf {f(x) \approx f(a)+f^{\prime}(a)(x-a)}[/imath]
is known as a linear approximation or tangent line approximation and is called a linearization [imath]\bf{\text{ of }f\text{ at }a}.[/imath]

rolexsyrup1212 see can do!​

[imath][/imath][imath][/imath]

 

[Steven G]

I believe in teaching in a way that gives a chance to be able to know the formula many years from now. One who understands calculus should remember one of the formulas for the derivative. Then they solve the formula for say f(a+h) as I did above. I don't walk around knowing the linear approximation formula but can derive it in my head at any time. I believe that is real teaching.

 

[Dr.Peterson]

I believe in teaching in a way that gives a chance to be able to know the formula many years from now. One who understands calculus should remember one of the formulas for the derivative. Then they solve the formula for say f(a+h) as I did above. I don't walk around knowing the linear approximation formula but can derive it in my head at any time. I believe that is real teaching.

Another way to think of it is that you are simply using the point-slope form of a line, with the derivative taken as the slope. (Of course, that form in turn is just a rearrangement of the definition of slope, which is equivalent to your formula for the approximate derivative.)

So there are lots of places to start, depending on what an individual finds most memorable.