Question

The equation of the tangent line to f(x) = √x at x = 49 is y =
Using this, we find our approximation for √{49.2} is..?

Answer 1

Did you find the equation for the tangent line yet?

Answer 2

no i don't know how to do this question

Answer 3

So a line is of the form:\[\Large\bf\sf y=mx+b\]The line tangent to our curve will have a special `slope`:\[\Large\bf\sf m=f'(49)\]The derivative evaluated at 49 gives us the slope of the line tangent to our curve at that point.

Answer 4

\[\Large\bf\sf f(x)=\sqrt x, \qquad\qquad f'(x)=?\]Do you understand how to find this derivative?

Answer 5

is it 7

Answer 6

You evaluated the `function` at x=49,\[\Large\bf\sf f(49)=\sqrt{49}\]\[\Large\bf\sf f(49)=7\]That gives us an ordered pair, \(\Large\bf\sf (49,\;7)\)
That's good, we'll need that information later.

Do you understand how to find the derivative of the function that was given?

Answer 7

nope sorry..

Answer 8

You can rewrite \(\Large\bf\sf \sqrt x\) as \(\Large\bf\sf x^{1/2}\) and then just apply the power rule from there.
But the square root shows up so often that it's worth memorizing,
so try to remember this one,\[\Large\bf\sf f(x)=\sqrt x,\qquad\qquad f'(x)=\frac{1}{2\sqrt x}\]Derivative of square root is one over 2 square roots.

Answer 9

Evaluating the derivative at 49 gives us the `slope` of our tangent line:
\[\Large\bf\sf f'(49)=\frac{1}{2\sqrt{49}}\]

Answer 10

So we have this tangent line,\[\Large\bf\sf y=mx+b\]We get the slope, `m`, from the derivative evaluated at x=49,\[\Large\bf\sf y=\frac{1}{14}x+b\]To find the y-intercept, `b`, we need to plug in the ordered pair that we came up with earlier.

Answer 11

ooooh so for b i would plug in 7?

Answer 12

We don't plug anything in directly for b, we're trying to solve for b.
We have an ordered pair that this line passes through,\[\Large\bf\sf (x,\;y)=(49,\;7)\]We plug those in for x and y and proceed to solve for b,\[\Large\bf\sf 7=\frac{1}{14}\cdot 49+b\]

Answer 13

So,\[\Large\bf\sf 7=\frac{49}{14}+b\]
Let's subtract 49/14 from each side,\[\Large\bf\sf 7-\frac{49}{14}=b\]Which tells us that our b is 3.5 I think.
My math look right?

Answer 14

Here is a graph to illustrate what we're doing.

Answer 15

i typed in 3.5 as my answer for y and it says it's wrong