I could use some help on this one. Calculate the slope of a line tangent to the curve of each of the functions $\displaystyle y = f(x)$ for the given point $\displaystyle P$.
$\displaystyle Y = x^2$ ; $\displaystyle P$ is $\displaystyle (2,4)$
I could use some help on this one. Calculate the slope of a line tangent to the curve of each of the functions $\displaystyle y = f(x)$ for the given point $\displaystyle P$.
$\displaystyle Y = x^2$ ; $\displaystyle P$ is $\displaystyle (2,4)$
First find the derivative $\displaystyle f'(x)=2x$...then you want to find the slope of the tangent line at the point $\displaystyle (2,4)$ and the derivative evaluated at a point x has the same slope as the tangent line at that point so the slope of the tangent is $\displaystyle f'(2)=2\cdot{2}=4$
If you haven't learned differntiation rules then we will go with the quotient difference...ok the slope of a curve is given by $\displaystyle f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$..so to get the devative of $\displaystyle f(x)=x^2$ before you know differentiation rules is to go through this thus the derivative of $\displaystyle x^2$ is $\displaystyle \lim_{h \to 0}\frac{(x+h)^2-x^2}{h}=\lim_{h \to 0}\frac{x^2+2xh+h^2-x^2}{h}=\lim_{h \to 0}\frac{2xh+h^2}{h}$$\displaystyle =\lim_{h \to 0}\frac{2xh}{h}+\lim_{h \to 0}\frac{h^2}{h}=2x+\lim_{h \to 0}h=2x$...so the slope at the point x is $\displaystyle 2\cdot{x}$...so the slope of the curve of $\displaystyle x^2$ at x=2 is $\displaystyle 2\cdot{2}=4$...so the slope of the tangent line at the point x=2 has the same slope as the curve so its slope is 4
all that work just to get $\displaystyle x^2$...but I'll tell you a little trick if $\displaystyle f(x)=x^{n}$...then the derivative of f(x) or f'(x) is $\displaystyle f'(x)=nx^{n-1}$...so to get the derivative of x²...just multiply the coefficient by the exponent and reduce the exponent by one..so it would be $\displaystyle f'(x)=2*x^{2-1}=2x^1=2x$
the limit stuff...hmm...thats something you just have to learn...here I think this will really help you its a great site..good luck! if you have any more questions just ask
Visual Calculus