Derivative help Calc 1

[aster111]

I have a homework assignment due tonight

Here are the two questions I still do not understand

1.) Write in slope intercept form the equation for the tangent line for the curve y=sin(x) at x=(pi/3).

What I do know is the f'(x)=cos(x).
Next, do I plug pi/3 into the derivative?? I'm lost at this point.

2.) Use the definition of the derivative to compute the derivative of f(x) = squareroot(x-3)

so.. f(x)=(x-3)^(1/2)
could I do f(x)=(1/2)(x-3)^(-1/2) ?
and where would I go after that?

 

[mmm4444bot]

If you are familiar with the Point-Slope Formula for writing the equation of a line, then you know that it's useful when you know the line's slope and the coordinates of one point on the line.

(Rearranging point-slope form to slope-intercept form is trivial.)

Given: y = sin(x)

The first derivative of y gives us the slope of the sine graph at any value of x.

Use the first derivative of y to calculate the slope of the sine graph where x = Pi/3.

Calculate the value of y where x = Pi/3.

Now you have everything you need (i.e., the slope of the line and the coordinates of one point on the line) to use the Point-Slope Formula to get the equation of the line tangent to the sine graph where x = Pi/3.

On the second exercise, you've used the Power Rule to determine the derivative. The instructions require that you use the definition of the derivative, instead.

In other words, I think that you're being asked to evaluate the limit of the difference quotient as h goes to zero.

 

[fasteddie65]

"2.) Use the definition of the derivative to compute the derivative of f(x) = squareroot(x-3)"

This is a standard problem in most calculus texts.

Let's do a similar problem: f(x) = ?(x - 1)

f(x + h) - f(x) = ?(x + h - 1) - ?(x - 1)

[f(x + h) - f(x)]/h = [?(x + h - 1) - ?(x - 1)]/h

Now rationalize the NUMERATOR:

[?(x + h - 1) - ?(x - 1)]/h • [?(x + h - 1) + ?(x - 1)]/[?(x + h - 1) + ?(x - 1)] =
(x + h - 1 - x + 1)/[?(x + h - 1) + ?(x - 1)]h =
h/[?(x + h - 1) + ?(x - 1)]h = 1/[?(x + h - 1) + ?(x - 1)]

Now, take the limit as h --> 0: 1/[2?(x - 1)]