Consider the quadratic function f ( x ) = − 4 x 2 + 32 x f\left(x\right)=-4x^2+32x f ( x ) = − 4 x 2 + 32 x . Calculate the slopes of the secant lines between the points ( x , f ( x ) ) \left(x,f\left(x\right)\right) ( x , f ( x ) ) and ( 4 , f ( 4 ) ) \left(4,f\left(4\right)\right) ( 4 , f ( 4 ) ) , for x = 3.5 , 3.9 , 3.99 , 3.999 x=3.5,3.9,3.99,3.999 x = 3.5 , 3.9 , 3.99 , 3.999 . Then, make a presumption on the value of the limit of the slopes of the secant lines between ( x , f ( x ) ) \left(x,f\left(x\right)\right) ( x , f ( x ) ) and ( 4 , f ( 4 ) ) \left(4,f\left(4\right)\right) ( 4 , f ( 4 ) ) as x x x gets closer to 4 4 4 .

2, 0.4, 0.04, 0.004; The limit is 0.

12, 0.8, 0.06, 0.003; The limit is 0.

6, 0.3, 0.03, 0.003; The limit is 6.

10, 0.5, 0.05, 0.005; The limit is 10.

Consider the quadratic function f(x)=−4x2+32xf\le...

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