Let f(x) = x2. f ( x) - f(y) x+ y a. Show that = f' for all x * y. x- y 2 b. Is this property true for f(x) = ax , where a is a nonzero real number? c. Give a geometrical interpretation of this property. d. Is this property true for f(x) = ax"?a. Begin by rewriting the left side of the equation. -w E] _ Substitute for f(x) and f(y) in the numerator. X ' y x - y E] Simplify. Now, rewrite the right side of the equation. Find f'(x). f'(x) = D Take the derivative. x + y Use the previous result to find f'[ 2 ] , x+y _ , _ X+Y , _ f[ 2 ] |:| Evaluate the derivative at 2 and Simplify. b. Is the property true for f(x) = ax2 ? Choose the correct answer below. a . nw , _ _ Ix+y A. No. The left Side, W, Includes a factor of a, but the right Side, f 2 , does not. fiX) - y) , X + y . . B. Yes. Both W and f 2 Simplify to the same result found above. . . , X+y . . f(X)-f(Y) C. No. The right Side, f 2 , Includes a factor of a, but the left Side, W, does not. fx -f f} D. Yes. Both % and f'[ x+y 2 simplify to the product of a and the result found above. c. Give a geometrical interpretation. Choose the correct answer below. The slope of the secant line between (x,f(x)) and (y,f(y)) is equal to the midpoint of the line segment from (x,f(x)) to (y,f(y)). The slope of the secant line between (x,f(x)) and (y,f(y)) is equal to the slope of the line between the origin and the midpoint ofthe line segment from (x,f(x)) to (y,f(y)). The slope of the secant line between (x,f(x)) and (y,f(y)) is equal to the slope of the line tangent to f(x) = x2 at the midpoint between x and y. 9.097? The slope of the secant line between (x,f(x)) and (y,f(y)) is equal to the sum of the slopes of the lines tangent to f(x) =X2 at x and at y. d. Is the property true for f(x) = axs? Choose the correct answer below. 3 if} A. No. When both sides are simplified, the right side differs from the left side by a factor of Z' ":22 B. Yes. When both sides are simplified, the result is the product of a and the square of the result found for f(x) =x2. 3 If} C. No. When both sides are simplified, the right side differs from the left side by both a factor of z and the coefcient on the xyterm. ":3 D. Yes. When both sides are simplified, the result is the product of 3a and the square of the result found for f(x) = X2. Find an equation of the line tangent to the following curve at the given point. y= -16x2 + 3 sin x; P(0,0) The equation for the tangent line is D