- WE'RE GIVEN Y = 2X SQUARED.
WE WANT TO FIND THE CHANGE IN Y, OR DELTA Y,
WHEN X = 2 AND DELTA X = 0.3.
DELTA Y REPRESENTS THE CHANGE IN Y ALONG THE FUNCTION,
IN THIS CASE FROM X = 2 TO X = 2.3.
WE KNOW X IS INCREASING, BECAUSE DELTA X IS POSITIVE.
IF DELTA X WAS NEGATIVE, THEN X WOULD BE DECREASING.
SO TO FIND DELTA Y WE'LL HAVE TO TAKE F OF X + DELTA X
AND SUBTRACT F OF X.
AND THEN WE'RE ASKED TO FIND DIFFERENTIAL Y
WHEN X = 2 AND DX = 0.3.
DIFFERENTIAL Y REPRESENTS THE CHANGE OF Y
ALONG THE TANGENT LINE, IN THIS CASE WHEN X = 2 TO X = 2.3.
SO DELTA Y REPRESENTS THE CHANGE OF Y ALONG THE FUNCTION,
AND DIFFERENTIAL Y REPRESENTS THE CHANGE OF Y
ALONG THE TANGENT LINE,
AND DIFFERENTIAL Y = F PRIME OF X x DX.
IF WE LOOK AT THIS GRAPHICALLY,
THE GIVEN FUNCTION IS GRAPHED HERE IN BLUE,
AND WE HAVE THE RED TANGENT LINE TO THE FUNCTION WHEN X = 2.
IF WE ZOOM IN ON THE POINT OF TANGENCY,
THE GRAPH WOULD LOOK LIKE THIS,
WHERE AGAIN, THE BLUE GRAPH IS THE FUNCTION
AND THE RED LINE IS THE TANGENT LINE.
SO WHEN WE FIND DELTA Y WE'RE FINDING THE CHANGE OF Y
ALONG THE BLUE FUNCTION,
AND WHEN WE FIND DIFFERENTIAL Y
WE'RE FINDING THE CHANGE OF Y ALONG THE TANGENT LINE.
BUT AS YOU CAN SEE FROM THE GRAPH,
THEY SHOULD BE VERY CLOSE TO EACH OTHER,
AND THEREFORE AT TIMES DIFFERENTIAL Y CAN BE USED
TO APPROXIMATE A CHANGE IN FUNCTION VALUES.
AND WE CAN SEE FROM THE GRAPH IT LOOKS LIKE DIFFERENTIAL Y
IS GOING TO BE A LITTLE BIT LESS THAN DELTA Y,
BECAUSE THE RED TANGENT LINE
IS A LITTLE BIT BELOW THE BLUE FUNCTION.
LET'S SEE HOW THIS WORKS OUT.
SO THE TRUE CHANGE IN Y,
OR DELTA Y, IS GOING TO BE EQUAL TO F OF X + DELTA X,
WHICH WOULD BE 2 + 0.3, - F OF X, WHICH IS F OF 2.
SO WE HAVE F OF 2.3 - F OF 2,
WHERE AGAIN, OUR FUNCTION IS Y = 2X SQUARED.
SO WE'D HAVE 2 x THE QUANTITY 2.3 SQUARED - 2 x 2 SQUARED.
LET'S GO AHEAD AND EVALUATE THIS ON THE CALCULATOR.
2 x 2.3 SQUARED - 2 x 2 SQUARED, WHICH OF COURSE
WOULD BE THE SAME AS 2 CUBED.
SO DELTA Y IS EXACTLY 2.58.
NOTICE TO FIND DIFFERENTIAL Y
WE'LL HAVE TO FIND THE DERIVATIVE FUNCTION.
SO IF Y = 2X SQUARED, NOTICE THAT Y PRIME,
WHICH WOULD BE F PRIME OF X IN OUR FORMULA, WOULD BE 4X.
SO AGAIN, FOR THE PURPOSE OF OUR FORMULA F PRIME OF X = 4X.
SO DIFFERENTIAL Y, THE CHANGE OF Y ALONG THE TANGENT LINE,
IS EQUAL TO F PRIME OF X, WHICH IS 2, x DIFFERENTIAL X OR DX,
WHICH IS 0.3.
WELL, F PRIME OF 2 WOULD BE 4 x 2 x 0.3, WHICH IS 2.4.
SO NOTICE, JUST AS WE EXPECTED,
DELTA Y IS A LITTLE BIT LARGER THAN DIFFERENTIAL Y.
IF WE GO BACK TO THE GRAPH JUST FOR A MOMENT,
NOTICE FROM 2 TO 2.3 DELTA Y
IS A LITTLE BIT LARGER THAN DIFFERENTIAL Y,
BECAUSE NOTICE AT 2.3
THE BLUE FUNCTION IS SLIGHTLY ABOVE THE RED TANGENT LINE.
I HOPE YOU FOUND THIS HELPFUL.