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- WE WANT TO DETERMINE F OF X GIVEN THAT F PRIME OF X = 3X
SQUARED AND F OF 2 = 7.
SO WE WANT TO DETERMINE A FUNCTION THAT HAS A DERIVATIVE
OF 3X SQUARED AND F OF 2 = 7.
WELL, DETERMINING A FUNCTION THAT HAS A KNOWN DERIVATIVE
SHOULD REMIND US OF THE ANTI-DERIVATIVE.
IN GENERAL, THE ANTI-DERIVATIVE,
OR INDEFINITE INTEGRAL OF F PRIME OF X WITH RESPECTS TO X,
IS GOING TO EQUAL F OF X + A CONSTANT C.
THIS FUNCTION HERE WILL HAVE A DERIVATIVE OF PRIME OF X.
SO FROM THE GIVEN INFORMATION,
THE INDEFINITE INTEGRAL OF 3X SQUARED WITH RESPECTS TO X
WOULD BE EQUAL TO OUR FUNCTION + A CONSTANT.
AND THEN WE CAN DETERMINE THE CONSTANT
BECAUSE WE'RE GIVEN F OF 2 = 7.
SO THIS ANTI-DERIVATIVE WOULD BE 3 x X TO THE 3rd
DIVIDED BY 3 + C.
THESE THREE SIMPLIFY OUT SO WE'RE LEFT WITH X CUBED + C.
SO WE KNOW THAT F OF X MUST BE EQUAL TO X CUBED + C.
THIS IS A FAMILY OF FUNCTIONS
THAT HAVE A DERIVATIVE OF 3X SQUARED,
BUT SINCE WE'RE ALSO GIVEN THAT F OF 2 = 7
WE CAN DETERMINE THE EXACT VALUE OF C
TO DETERMINE WHAT'S CALLED THE PARTICULAR SOLUTION
TO THIS PROBLEM.
SO F OF 2 = 2 CUBED + C = 7.
SO WE HAVE 8 + C = 7 SUBTRACTING 8 ON BOTH SIDES WE HAVE C = -1,
WHICH MEANS THE PARTICULAR SOLUTION,
OR THE EXACT FUNCTION THAT SATISFIES THESE CONDITIONS,
WOULD BE X CUBED - 1.
LET'S TAKE A LOOK AT THIS GRAPHICALLY.
WHAT WE'RE SEEING HERE IS THE GRAPH OF SEVERAL FUNCTIONS
THAT HAVE A DERIVATIVE OF 3X SQUARED.
SO ALL THESE FUNCTIONS ARE A MEMBER OF THE FAMILY
OF F OF X = X CUBED + C.
BUT SINCE WE WERE GIVEN THAT F OF 2 = 7,
WHICH MEANS WHEN X = 2 Y = 7,
THIS POINT MUST BE ON THE EXACT FUNCTION THAT WE'RE LOOKING FOR,
WHICH IS THIS POINT HERE.
AND NOTICE THAT THE ONLY FUNCTION THAT CONTAINS
THIS POINT IS F OF X = X CUBED - 1,
WHICH IS THE PARTICULAR SOLUTION
THAT SATISFIES THE GIVEN CONDITIONS.
I HOPE YOU FOUND THIS HELPFUL.