We would need to calculate the derivative of this function to do so. The derivative of is simply So at any give point x, the slope of the curve at that point is 2x. For example, at the point 5,25 the slope of the tangent line to the curve is or But at the point 0,0 the slope of the tangent line to the curve is or 0, because the tangent line to the curve at 0,0 is horizontal. Short answer: The slope of the tangent line at any point on the curve is given by which is known as the derivative of y wth respect to x.
I hope this helps! Keep practicing! Paypal is always accepted for detailed assistance with single problems, send contributions to: neatmath yahoo. Assistance for single problems can be done via email with scans or photos to help illustrate concepts. Thank you and good luck with all your future studies! In other words, given a "word problem" modelling something in the real world, or an actual real-world linear model, what do the slope and intercept of the modelling equation stand for, in practical terms?
Back when we were first graphing straight lines, we saw that the slope of a given line measures how much the value of y changes for every so much that the value of x changes. For instance, consider this line:. This means that, starting at any point on this line, we can get to another point on the line by going up 3 units and then going to the right 5 units.
But and this is the useful thing we could also view this slope as a fraction over 1 ; namely:. This tells us, in practical terms, that, for every one unit that the x -variable increases that is, moves over to the right , the y -variable increases that is, goes up by three-fifths of a unit.
While this doesn't necessarily graph as easily as "three up and five over", it can be a more useful way of viewing things when we're doing word problems or considering real-world models.
Slope: Very often, linear-equation word problems deal with changes over the course of time; the equations will deal with how much something represented by the value on the vertical axis changes as time represented on the horizontal axis passes. An exercise might, say, talk about how the population grows, year on year, in a certain city, assuming that the population increases by a certain fixed amount every year.
For every year that passes that is, for every increase of 1 along the horizontal axis , the population would increase that is, move up along the vertical axis by that fixed amount. For a time-based exercise, this will be the value when you started taking your reading or when you started tracking the time and its related changes. In the example from above, the y -intercept would be the population when the sociologists started keeping track of the population.
Advisory: "When you started keeping track" is not the same as "when whatever it is that you're measuring started". Using the example above, your population-growth model might be very accurate for the years through , but the city whose population is being measured might have been founded way back in What is the slope? This value tells me that, for every increase of 1 in my input variable t that is, for every increase of one year , the value of my output variable y will increase by 0.
The slope tells me that, every year, the average lifespan of American women increased by 0. The intercept value tells me that, in when they started counting , the average lifespan of an American woman was 73 years.
This value tells me that, for every increase by 1 in my input variable t , I get a decrease of 32 in my output variable v. The slope tells me that, for every second that passes, the speed of the ball decreases by 32 feet per second. Also, by the way, the velocity will eventually become zero when the ball reaches the peak of its arc , and will then become negative when gravity takes over and pulls the ball back down to the ground.
The exercise defines v as measuring the velocity of the ball. The intercept value tells me that, when the ball was released, it was launched upward at a speed of feet per second.
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