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: \( m_{sec} = \frac{f(x) - f(2)}{x - 2} \).
Calculate \( f(2) \) by substituting \( x = 2 \) into the function \( f(x) = -16x^2 + 64x \). This will give you the y-coordinate of the point (2, f(2)).
Substitute \( f(x) = -16x^2 + 64x \) and \( f(2) \) into the difference quotient formula: \( m_{sec} = \frac{-16x^2 + 64x - f(2)}{x - 2} \).
Simplify the expression for \( m_{sec} \) by performing polynomial division or factoring, if possible, to eliminate the \( x - 2 \) in the denominator.
Finally, make a conjecture about the limit of \( m_{sec} \) as \( x \) approaches 2. This involves evaluating the simplified expression for \( m_{sec} \) as \( x \to 2 \), which will give you the slope of the tangent line at \( x = 2 \).
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