Answer:
Work Shown:
Note that (h,k) = (2,-3) is the center and r = 6 is the radius. The last line is not expanded out using the FOIL rule because its common practice to leave it in form. This form allows the reader to quickly determine the center and radius (rather than have to do a lot of algebraic steps before hand).
Answer:
The tangent line to the given curve at the given point is .
Step-by-step explanation:
To find the slope of the tangent line we to compute the derivative of and then evaluate it for
.
Differentiate the equation.
Differentiate both sides.
Sum/Difference rule applied:
Constant multiple rule applied:
Applied power rule:
Simplifying and apply constant rule:
Simplify.
Evaluate y' for x=4:
is the slope of the tangent line.
Point slope form of a line is:
where is the slope and
is a point on the line.
Insert 9 for and (4,10) for
:
The intended form is which means we are going need to distribute and solve for
.
Distribute:
Add 10 on both sides:
The tangent line to the given curve at the given point is .
------------Formal Definition of Derivative----------------
The following limit will give us the derivative of the function at
(the slope of the tangent line at
):
We are given f(4)=10.
Let's see if we can factor the top so we can cancel a pair of common factors from top and bottom to get rid of the x-4 on bottom:
Let's check this with FOIL:
First:
Outer:
Inner:
Last:
---------------------------------Add!
So the numerator and the denominator do contain a common factor.
This means we have this so far in the simplifying of the above limit:
Now we get to replace x with 4 since we have no division by 0 to worry about:
2(4)+1=8+1=9.