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Numerade Educator

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Problem 1 Easy Difficulty

A curve has equation $ y = f(x) $.
(a) Write an expression for the slope of the secant line through the points $ P(3, f(3)) $ and $ Q(x, f(x)) $.
(b) Write an expression for the slope of the tangent line at $ P $.

Answer

a) $m=\frac{f(x)-f(3)}{x-3}$
b) $\lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3}$

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Daniel N.

July 12, 2021

Suppose that f(x) is a function and that for any given ? > 0, the condition 0 < |x ? 2| < 3 4 ? guarantees that |f(x) ? 5| < ?. (a) What limit is described by this statement?

Video Transcript

for this problem, given a function Y equals f of X. We need to find a slope of the second line passing through the point P&Q. With p having coordinates three up of three in Q with coordinates X. F of X. Also, we need to find the slope of the tangent line at P. I consider this arbitrary graph of the function with points P and Q in it, the line passing through points B and Q. Is the second line drawn here. Know that the slope of this line is just the change and why over the change in X. And so the slope of the second line through points P and Q is F of X -F. L. three over x minus three. For part B. We draw the tangent line at P know that as a cube gets closer to P, the second lines form slowly follows the form of the tangent line at P. So the limiting value of the slope of the second line through P and Q is equal to the slope of the tangent line at P. That means the slope of the tangent line at P. This is equal to the limit of f of x minus ever. Three Over X -3. As X approaches three