Question

Can I get some Math help?

Answer 1

Any idea on how to do it?

Answer 2

calculus

Answer 3

when you're calculating limits this way, simply evaluate what's in the expression

so when it says f(x0 + h), plug x0 + h into the function f(x) to get

3sqrt(x0 + h), etc. and continue with f(x0)

when you expand/simplify the expression, you should be able to cancel out h from the denominator, which lets you plug in h = 0 and get the final limit

Answer 4

would it be undefined?

Answer 5

no, there's definitely a finite limit on this one

\[{ \lim_{h \rightarrow 0} }{ \frac{ f(x_{0}+h)-f(x_{0}) }h }\]
plugging in f(x) = 3sqrt(x)

\[{ \lim_{h \rightarrow 0} }{ \frac{3\sqrt{x_{0}+h}-3\sqrt{x_{0}} }h }\]

plugging in x0 = 4

\[{ \lim_{h \rightarrow 0} }{ \frac{3\sqrt{4+h}-6 }h }\]

Answer 6

to make the arithmetic simpler, factor out the 3 from the numerator to get

\[{ \lim_{h \rightarrow 0} }{ \frac{ 3 }{ h }(\sqrt{4+h}-2) }\]

rationalize the numerator by multiplying the expression by the conjugate of sqrt(4+h)-2, which is sqrt(4+h) - 2

\[{ \lim_{h \rightarrow 0} }{ \frac{ 3 }{ h }(\sqrt{4+h}-2) }\frac{ \sqrt{4+h}+2 }{ \sqrt{4+h}+2 }\]\]

simply the numerator using difference of squares. if done correctly, you'll notice that the numerator loses all of its radicals, and you're left with an h you can cancel out from the numerator and denominator. from there, you can plug in h = 0 and get your final limit.

Answer 7

thank you I got it 3/4

Answer 8

could you help me with one more problem please

Answer 9

I'll give it a try sure

Answer 10

I'm sure there's a smarter way to do this, but

you could calculate the slope between points (5,-8) and (2,4) using the slope formula, then use the point-slope equation to get the full equation of that line in y = mx + b form

from there you could re-write 20y - kx = 4 as y = (k/20)x - (1/5) and then set the slopes equal to each other since they're parallel

Answer 11

k would equal -82?

Answer 12

hmm I got something a bit different

slope formula: \[\frac{ (y_{2}-y_{1}) }{ (x_{2}-x_{1} }=\frac{ (4-(-8) }{ (2-5) }=-4\]

so the slope is -4. since both lines are parallel, both slopes are -4

so if y = (k/20)x - (1/5) has a slope of -4, then (k/20) = -4, should be straightforward what k is.