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Determine the coordinates of the midpoint of the line segment joining the points (a) $(-2,5)$ and $(6,8),(b)(3,-7)$ and $(-12,15).$

(a) $(2,13 / 2)$(b) $(-9 / 2,4)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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this problem. We want to find the midpoint of two different line segments, and we have each one just number these one and two. Each pair represents the end point of a line segment. So what is the midpoint directly between each of these pairs? Well, in order to do that, let's just take a moment to sketch. What? Something like this might look like I'm not going to use these particular numbers. I'm just going to pick two points, right? And I'm going to draw the line between them now, just visually. I can tell the midpoint is going to be somewhere about there if I've got these as my endpoints, but let's think of what we have. Okay? I'm gonna do blue here. I'm going on the X axis. I'm going from Coordinate one to coordinate to. I want to go directly in the middle. How do you find the middle of two numbers? That's the average. So in order to find my ex, I'm going to have to average my two x coordinates. That's the X coordinate of that midpoint by the same token, look what's happening for my y coordinate. I'm going from here to here on the Y axis and I want the midpoint. How do you find the midpoint again? It's the average. So the X coordinate is the average of my two. X is the Y. Coordinate is the average of my two wise. So let's go over here and see how this works for our pears. Let's take care. Number one. Well, I have negative two and six, so the average is going to be I'm going to add them and divide by two. That's how we find an average. For my wise, I add them, and I divide by two. So what does that give me for a midpoint that could be four divided by two or two and then end up with 13 halves? How about my second pair? Start with my exes. I add my exes. Divide by two, and I add my wise and again divide by two. So that gives me a midpoint of negative nine haves, and that's eight halves or four. So those are my two mid points

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