[ MUSIC ]
- WELCOME TO APPLICATIONS USING DIFFERENTIATION TECHNIQUES.
WE'LL COVER THE CONSTANT RULE, POWER RULE,
CONSTANT MULTIPLE RULE,
AND THE SUM OF DIFFERENCE RULE WITH AN APPLICATIONS.
THE GOALS WILL BE TO FIND THE EQUATION OF A TANGENT LINE
AND FIND THE POINTS ON A FUNCTION
THAT HAVE A HORIZONTAL TANGENT LINE.
LET'S TAKE A LOOK AT AN EXAMPLE.
FIND THE EQUATION OF THE TANGENT LINE AT THE POINT (4,14)
TO THE FUNCTION Y = X SQUARED - THE SQUARE ROOT OF X.
ON OUR PREVIOUS VIDEO WE DISCUSSED
THAT WE CAN'T USE THE POWER RULE WHEN IT'S IN RADICAL FORM.
SO THE FIRST STEP WILL BE TO REWRITE THIS RATIONAL EXPONENTS.
SO WE'LL HAVE Y = X SQUARED - X TO THE POWER OF 1/2.
NOW, IF WE WANT TO FIND THE EQUATION OF THE TANGENT LINE,
OF COURSE WE'RE GOING TO HAVE TO FIND THE SLOPE OF THAT LINE.
SO IF WE FIND THE DERIVATIVE AND EVALUATE IT AT X = 4,
WE'LL FIND THE SLOPE OF THAT TANGENT LINE.
LET'S TRY IT.
Y PRIME IS EQUAL TO THE DERIVATIVE OF X SQUARED,
WHICH WILL BE 2X - THE DERIVATIVE OF X TO THE 1/2.
THAT WOULD BE 1/2 x X TO THE POWER OF 1/2 - 1,
WHICH WOULD BE -1/2.
NOW WE'LL SIMPLIFY THIS.
SIMPLIFYING THIS TERM,
IF I MOVE THIS DENOMINATOR I'LL HAVE X TO THE +1/2 POWER,
WHICH IS THE SAME AS THE SQUARE ROOT OF X.
NOW I NEED TO EVALUATE THIS WHEN X IS 4.
SO WE'D HAVE 2 x 4 - 1 DIVIDED BY 2 x THE SQUARE ROOT OF 4.
SO Y PRIME OF 4 WILL BE EQUAL TO 8 - SQUARE ROOT OF 4 IS 2
x THIS 2 WOULD GIVE US A - 1/4 OR - 1/4,
WHICH WOULD EQUAL 31/4 OR 7.75.
NOW, THIS WILL BE THE SLOPE OF OUR TANGENT LINE.
SO NOW WE NEED TO FIND THE EQUATION OF THE TANGENT LINE
AND WE'LL USE POINT SLOPE FORM.
JUST IN CASE YOU DON'T REMEMBER THAT, HERE IT IS.
AGAIN, OUR SLOPE IS HERE, OUR POINT IS HERE.
THIS WOULD GIVE US Y - 14 IS EQUAL TO 7.75
x THE QUANTITY X - 4.
WE'LL SOLVE THIS FOR Y.
ADD 14 TO BOTH SIDES AND WE HAVE Y IS EQUAL TO 7.75X - 17.
OKAY. LET'S VERIFY OUR WORK BY LOOKING AT A GRAPH.
HERE WE HAVE OUR FUNCTION.
HERE WE HAVE THE GRAPH OF OUR TANGENT LINE.
HERE'S THE POINT THAT WAS REFERRED TO IN THE PROBLEM,
(4,14).
AND WE CAN SEE THAT OUR LINE IS TANGENT AT THAT POINT,
WHICH VERIFIES OUR WORK.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
FIND THE POINTS ON THE FUNCTION
FOR WHICH THE TANGENT LINE WOULD BE HORIZONTAL.
WELL, ALL HORIZONTAL LINES HAVE A SLOPE OF ZERO,
SO ON THIS PROBLEM WE NEED TO FIND
WHERE THE DERIVATIVE WOULD BE EQUAL TO ZERO.
REMEMBER THE DERIVATIVE GIVES US THE SLOPE OF THE TANGENT LINES.
SO LET'S FIND OUR DERIVATIVE AND SET IT EQUAL TO ZERO.
SO DERIVATIVE OF -1/3X CUBE WOULD BE -1/3
x THE DERIVATIVE OF X CUBE,
WHICH WOULD BE 3X SQUARED,
+ 6 x THE DERIVATIVE OF X SQUARED THAT WOULD BE 2X.
- 11 X THE DERIVATIVE OF X THAT WOULD BE 1,
AND THEN - DERIVATIVE OF 50 THAT WOULD BE 0.
LET'S SIMPLIFY HERE AND SET IT EQUAL TO ZERO.
THOSE THREE SIMPLIFY.
THIS WOULD LEAVE US WITH -X SQUARED + 12X - 11.
AGAIN, WE WANT TO KNOW WHEN THIS IS EQUAL TO ZERO.
THAT'S WHERE THE TANGENT LINES WOULD BE HORIZONTAL.
LET'S FACTOR OUT THE NEGATIVE.
NOTICE IT CHANGES THE SIGN OF ALL OF THE TERMS.
AND THIS DOES FACTOR.
THE FACTORS OF +11 AT THE -12 THAT'D BE -11 AND A -1.
SO THE SOLUTIONS TO THIS WOULD BE X = 11 AND X = 1.
NOW, THESE ARE THE X COORDINATES,
BUT THEY ASK FOR THE POINTS,
SO WE ALSO HAVE TO FIND THE Y COORDINATES.
SO WE NEED TO FIND F OF 11.
THAT WOULD GIVE US THE Y COORDINATE
WHEN X IS 11 AND F OF 1.
REMEMBER WE'RE GOING TO SUB THESE VALUES
BACK INTO THE ORIGINAL FUNCTION HERE NOT THE DERIVATIVE.
SAVE TIME, I'VE ALREADY DONE THAT.
F OF 11 IS EQUAL TO 111.3 REPEATING.
F OF 1 IS EQUAL TO -55.3 REPEATING.
SO THE COORDINATES OF THE POINTS
WHERE THE TANGENT LINES WOULD BE HORIZONTAL
ARE (11,111.3 REPEATING), AND (1,-55.3 REPEATING).
AGAIN, LET'S VERIFY IT WITH A GRAPH.
IN BLUE WE HAVE OUR FUNCTION.
IN RED YOU CAN SEE OUR TWO HORIZONTAL TANGENT LINES.
THE POINTS OF TANGENCY ARE THE POINTS
THAT WE FOUND HERE AND HERE.
I WOULD ALSO LIKE YOU TO NOTICE THE RELATIVE EXTREMA OCCUR
WHERE THE DERIVATIVE IS EQUAL TO ZERO.
HERE WE HAVE A RELATIVE MINIMUM,
AND HERE WE HAVE A RELATIVE MAXIMUM.
WE'LL COME BACK TO THIS IDEA.
THANK YOU FOR WATCHING. I HOPE THAT HELPS.
HAVE A WONDERFUL DAY.