How To Use Tangent To Find An Angle
This is a solved case from Calculus Made Easy (you can download information technology from the link), I have the solution only can't sympathise it.
- Problem one: How did author find the bending ?
- Problem two: Author insists that it is tangent, but graph shows information technology's secant.
- Trouble three: Page 86, writer writes: Now, when two curves meet, the intersection existence a bespeak common to both curves, its coordinates must satisfy the equation of each one of the ii curves; He is talking about intersection of 2 curves here but equations he equates are of tangent and one curve. Information technology is disruptive.
Argument: Observe the slope of the tangent to the bend $y = \frac{1}{2x} + 3$ at $x = -1$. Detect the angle this tangent makes with second bend $y = 2x^2 + 2$.
SOLUTION: at $x = -1,$ Slope of tangent to bend = Slope of the curve.
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$\frac{dy}{dx}$ = $\frac{-ane}{2x^2}$, putting 10 = -1, $\frac{dy}{dx}$ = $\frac{-ane}{ii}$.
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Tangent is a straight line, $y = ax + b$, which means $\frac{dy}{dx}$ = $a$ = $-1/2$
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For beginning bend, $x = -1$ ways $y = \frac{5}{2}$.
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Nosotros got values for $x$, $y$ and $a$. Since tangent touches and curve touch at one point, co-ordinates of that point will be same for both tangent and curve must be same. To observe $b$ nosotros put put all these values in equation for tangent $y = ax + b$, which gives us $b = 2$. hence equation of tangent is $y = \frac{-ten}{2} +2$.
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From (4). equation for beginning curve = equation for tangent . $2x^2 + 2 = \frac{-10}{ii} + 2$. Solving this we get $x = 0$ or $x = \frac14$
Now we will go to 2nd curve
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$y = 2x^two +2$, $\frac{dy}{dx} = 4x$ and for $10 = \frac{-1}{4}$, $\frac{dy}{dx} = -1$. Since tan(45) = 1. Hence the slope for this 2nd curve is downwards.
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Slope for the tangent is -1/2, hence that is downwardly also with $tan(angle) = \frac{i}{two}$.
I do not empathise from here onwards
it slopes downwards to the right and makes with the horizontal an angle φ such that tan φ = 12 ; that is, an angle of 26 degrees 34 minutes. Information technology follows that at the outset point the curve cuts the straight line at an angle of 26 degrees 34 minutes, while at the second it cuts it at an angle of 45 degrees − 26 degrees 34 minutes = 18 degrees 26 minutes. Hence the answer is eighteen degrees 26 minutes
Source: https://math.stackexchange.com/questions/2876006/calculus-made-easy-finding-angle-of-a-tangent
Posted by: sowellholed1992.blogspot.com
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