Given the function f(x)=−16x2+64x, complete the following. <IMAGE> Make a conjecture about the value of the limit of the slopes of the secant lines that pass through (x,f(x)) and (2,f(2)) as x approaches 2.
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First, understand that the slope of the secant line between two points (x, f(x)) and (2, f(2)) on the curve is given by the difference quotient: .
Calculate by substituting into the function . This will give you the y-coordinate of the point (2, f(2)).
Substitute and into the difference quotient formula: .
Simplify the expression for by performing polynomial division or factoring, if possible, to eliminate the in the denominator.
Finally, make a conjecture about the limit of as approaches 2. This involves evaluating the simplified expression for as , which will give you the slope of the tangent line at .
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