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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 $.

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02:10

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

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?

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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A curve has equation $ y =…

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A curve has equation $y=f(…

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Explain how to find the sl…

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a. Find the slope of the t…

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(a) Use Definition (3.1) t…

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

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