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Horizontal and Vertical Tangents of Parametric Functions | tangent line คือ

Horizontal and Vertical Tangents of Parametric Functions


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The title says it all.

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Horizontal and Vertical Tangents of Parametric Functions

11.3 The Tangent Line Problem (intro)


11.3 The Tangent Line Problem (intro)

TANGENT LINES AND CIRCLES EXPLAINED!


Let’s explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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Understand and apply theorems about circles.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

What is a tangent line to a circle?
Why is a tagent line and a radius perpendicular?
How many times does a tangent line cross a circle?

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TANGENT LINES AND CIRCLES EXPLAINED!

What is the tangent line?


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Tangent lines are absolutely critical to calculus; you can’t get through Calc 1 without them! In this video, we’re talking all about the tangent line: what it is, how to find it, and where to look for vertical and horizontal tangent lines.

The tangent line is useful because it allows us to find the slope of a curved function at a particular point on the curve. We learned a long, long time ago in a math class far, far away that we could find the slope of a line, but we’ve never learned how to find the slope of a curved function. Since the slope of a curved function is always changing, the best we can do is find the slope of the curved function at one particular point on the function. And to do this, we actually don’t look at the function at all. Instead, we look at the tangent line to the curve that passes through the particular point we’re interested in, and we find the slope of the line instead.

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To find the slope of the curve, all we have to do is take the derivative of the curve (because the derivative represents the slope), and then find the line with the correct slope that passes through the point of tangency. That’ll give us the tangent line, and the tangent line will have the same slope as the slope of the curve at the point of tangency.

We’ll also look at where to find vertical tangent lines, and where to find horizontal tangent lines, since that’s something you’ll be asked to do often. Horizontal tangent lines exist where the derivative of the function is equal to 0, and vertical tangent lines exist where the derivative of the function is undefined.

0:24 // The definition of the tangent line
1:16 // How to find the equation of the tangent line
3:10 // Where the tangent line is horizontal and vertical

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What is the tangent line?

How to Find the Equation of a Tangent Line with Derivatives (NancyPi)


MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.

For HOW TO FIND DERIVATIVES, jump to: https://youtu.be/QqF3i1pnyzU

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What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.

Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:
There are four main steps to find the equation of a tangent line with derivatives:

1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x, by the power rule.

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2) PLUG IN XVALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given xvalue into the derivative of f(x). The value you get is the slope of the tangent line, m.

3) FIND YVALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the yvalue of the point. You get this by plugging the given xvalue into the original equation to find the corresponding yvalue. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a yvalue of 7. Now that you have both the x and y coordinates, this means that the full point is (4,7).

4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINTSLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point slope equation of a line. The point slope form is x x1 = m (y y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.

You can also rearrange the equation you get so that it is in “y equals” or slope intercept form, if you want.

You may need to find the derivative with other derivative rules, such as the quotient rule or chain rule. Even if you need to use other rules when taking derivatives, or even if you need to differentiate something like a trigonometry function (ex. secant) or a fraction form with a numerator and denominator (rational function), the four main steps to find the equation of the tangent line will be the same.

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How to Find the Equation of a Tangent Line with Derivatives (NancyPi)

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