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- LETS TAKE A LOOK AT ONE MORE EXAMPLE.
A COMPANY FOUND THE DEMAND FOR ITS PRODUCT TO BE D OF X
EQUAL TO THE SQUARE ROOT OF 600 - X.
FIND THE ELASTICITY FUNCTION.
I'LL GO AHEAD AND WRITE THAT OUT AGAIN.
OKAY, ON THIS PROBLEM WE HAVE TO FIND THE DERIVATIVE OF THIS,
WHICH IS GOING TO REQUIRE THE CHAIN RULE.
SO LET'S GO AHEAD AND REWRITE THIS SO WE CAN WORK WITH IT,
MEANING I'M GOING TO WRITE D OF X = 600 - X
RAISED TO THE 1/2 POWER.
REMEMBER THE SQUARE ROOT IS THE SAME AS THE 1/2 POWER,
THEREFORE FOR OUR EARLY ELASTICITY FUNCTION--
OKAY, THE DERIVATIVE OF THIS WOULD BE WHAT 1/2 x 600 - X
RAISED TO THE -1/2 POWER.
REMEMBER x -1/OUR DEMAND FUNCTION.
AND I'M GOING TO GO AHEAD AND WRITE IT
USING A RATIONAL EXPONENT.
OKAY, NOW I DIDN'T LEAVE ENOUGH ROOM
SO I'M GOING TO GO AHEAD AND CLEAN THIS UP
AND WRITE IT DOWN BELOW HERE.
NOW, THERE'S SEVERAL THINGS GOING ON HERE.
THE FIRST THING I NOTICE, THERE'S A -X x -1,
SO I'LL HAVE A X IN THE NUMERATOR.
NOW, THIS TWO WOULD END UP IN THE DENOMINATOR.
AND REMEMBER THAT IF I TOOK THIS,
SINCE IT'S ALL BEING MULTIPLIED TOGETHER,
IF I TOOK THIS 600 - X RAISED TO THE -1/2 POWER
AND MOVED IT DOWN TO THE DENOMINATOR,
REMEMBER IT'S GOING TO MAKE THIS EXPONENT INTO A 1/2.
SO THEREFORE, I WOULD ADD THE EXPONENT HERE,
WHICH WOULD RESULT IN THE DENOMINATOR BEING 600 - X
TO THE 1st POWER.
OKAY, LET'S GO AHEAD AND LOOK AT PART B AGAIN.
THIS IS THE ANSWER FOR PART A.
FIND THE ELASTICITY AT PRICE X EQUALS 98.
SO WE NEED TO FIND E OF 98.
SO WE HAVE 98 IN THE NUMERATOR DIVIDED BY 2 x--
NOW 600 - 98 WOULD BE 502.
AND THIS IS GOING TO EQUAL 0.098.
AGAIN, THIS IS VERY INELASTIC.
AN INCREASE IN PRICE WILL BRING AN INCREASE IN REVENUE.
LASTLY, WE WANT TO FIND THE PRICE, OR VALUE OF X,
WHICH REVENUE IS A MAXIMUM.
REMEMBER THAT THAT'S WHEN THE ELASTICITY IS EQUAL TO ONE.
SO WHAT WE'RE GOING TO DO NOW IS TAKE OUR ELASTICITY FUNCTION,
SET IT EQUAL TO ONE AND SOLVE.
I LIKE TO DO CROSS PRODUCTS,
SO I'M GOING TO WRITE THIS AS 1/1,
WHICH MEANS THAT IF I DO MY CROSS PRODUCTS
X x 1 = 2 x THE QUANTITY 600 - X x 1.
DISTRIBUTE THE 2, ADD 2X TO BOTH SIDES, DIVIDE BY 3, X = 400.
THE PRICE OF $400 WILL MAXIMIZE OUR REVENUE.
SO THE PRICE OF $98 IS WAY TOO LOW
IF WE WANT TO MAXIMIZE REVENUE.
I HOPE THAT HELPS EXPLAIN ELASTICITY OF DEMAND
AND SOME OF ITS APPLICATIONS.
THANK YOU FOR TUNING IN AND HAVE A GOOD DAY.