How do you find the equation of the line tangent to the graph of #Y= (X-3) / (X-4)# at (5,2)?
2 Answers
Explanation:
The challenge, of course, is to find the slope of the tangent line. Once we know that the slope of the tangent line at that point is
There are shortcut methods, simplified rules and properties for finding slopes of tangent lines.
If you haven't learned them yet, then you are probably working with one of the definitions for the slope of the line tangent to the graph of
(If one of these limit exist, then both do and they are equal.)
For this problem we have
So we want
If we try to evaluate the limit by substitution, we get the indeterminate form
# = lim_(xrarr5)(((x-3)-2(x-4))/(x-4))/(x-5) # #" "# form is#0/0#
# = lim_(xrarr5)(-x+5)/((x-4)(x-5))# #" "# form is#0/0#
# = lim_(xrarr5)(-1(x-5))/((x-4)(x-5))# #" "# form is#0/0#
# = lim_(xrarr5)(-1)/(x-4) = -1/1 = -1#
Now that we have slope
Using the quotient rule and other rules:
At
The answer is
Explanation:
To find the equation of the tangent line to any graph, these rules must apply:
- Find the derivative of the function for any
#x# . Remember the idea of secant lines coming together to one point (the instantaneous rate of change or the tangent line). - Given the point, insert the
#x# to find the derivative of the function at that point (which will give you the slope of the tangent line). - Use point-slope form to find the equation of the tangent line.
First off, the equation
We can simplify it to when we can use the Power Rule and the Chain Rule to solve for the derived function of
So
Now we need the to include the point
Finally we can use point-slope form to find the equation of the tangent line:
For the function
or
You can always check to see if it works with a graphing calculator, but the key thing is to understand why you made a mistake for better confidence.