It'd be 32 degrees F warmer
13-(-19)
=32
Answer: 212998.58
Step-by-step explanation:
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if the price increases 6% every year for 7 years you would have to multiply 7 by 6 which is 42 so you have to add 42% to 149,999 which is 212998.58.
The slope of the tangent line to the curve at (8, 2) is given by the derivative
at that point. By the chain rule,
![\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{dx} = \dfrac{\frac{dy}{dt}}{\frac{dx}{dt}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cdfrac%7Bdy%7D%7Bdt%7D%20%5Ctimes%20%5Cdfrac%7Bdt%7D%7Bdx%7D%20%3D%20%5Cdfrac%7B%5Cfrac%7Bdy%7D%7Bdt%7D%7D%7B%5Cfrac%7Bdx%7D%7Bdt%7D%7D)
Differentiate the given parametric equations with respect to
:
![x = 4t \implies \dfrac{dx}{dt} = 4](https://tex.z-dn.net/?f=x%20%3D%204t%20%5Cimplies%20%5Cdfrac%7Bdx%7D%7Bdt%7D%20%3D%204)
![y = \dfrac4t \implies \dfrac{dy}{dt} = -\dfrac4{t^2}](https://tex.z-dn.net/?f=y%20%3D%20%5Cdfrac4t%20%5Cimplies%20%5Cdfrac%7Bdy%7D%7Bdt%7D%20%3D%20-%5Cdfrac4%7Bt%5E2%7D)
Then
![\dfrac{dy}{dx} = \dfrac{-\frac4{t^2}}4 = -\dfrac1{t^2}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cdfrac%7B-%5Cfrac4%7Bt%5E2%7D%7D4%20%3D%20-%5Cdfrac1%7Bt%5E2%7D)
We have
and
when
, so the slope at the given point is
.
The normal line to the same point is perpendicular to the tangent line, so its slope is +4. Then using the point-slope formula for a line, the normal line has equation
![y - 2 = 4 (x - 8) \implies \boxed{y = 4x - 30}](https://tex.z-dn.net/?f=y%20-%202%20%3D%204%20%28x%20-%208%29%20%5Cimplies%20%5Cboxed%7By%20%3D%204x%20-%2030%7D)
Alternatively, we can eliminate the parameter and express
explicitly in terms of
:
![x = 4t \implies t = \dfrac x4 \implies y = \dfrac4t = \dfrac4{\frac x4} = \dfrac{16}x](https://tex.z-dn.net/?f=x%20%3D%204t%20%5Cimplies%20t%20%3D%20%5Cdfrac%20x4%20%5Cimplies%20y%20%3D%20%5Cdfrac4t%20%3D%20%5Cdfrac4%7B%5Cfrac%20x4%7D%20%3D%20%5Cdfrac%7B16%7Dx)
Then the slope of the tangent line is
![\dfrac{dy}{dx} = -\dfrac{16}{x^2}](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdx%7D%20%3D%20-%5Cdfrac%7B16%7D%7Bx%5E2%7D)
At
, the slope is again
, so the normal has slope +4, and so on.
Answer:
See explanation
Step-by-step explanation:
The range of the cosine function is ![-1\le \cos x\le 1](https://tex.z-dn.net/?f=-1%5Cle%20%5Ccos%20x%5Cle%201)
Therefore
is not defined.
Assuming your question is rather ![arc \cos (-\frac{\sqrt{2}}{2})](https://tex.z-dn.net/?f=arc%20%5Ccos%20%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29)
Then
in the second quadrant.
![arc \cos (-\frac{\sqrt{2}}{2})=\pi-\frac{\pi}{4}](https://tex.z-dn.net/?f=arc%20%5Ccos%20%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B4%7D)
![arc \cos (-\frac{\sqrt{2}}{2})=\frac{3\pi}{4}](https://tex.z-dn.net/?f=arc%20%5Ccos%20%28-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D)