4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function...
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Transcript of 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function...
![Page 1: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/1.jpg)
4.1 Linear ApproximationsFri Oct 16
Do NowFind the equation of the tangent
line of each function at 1) Y = sinx
2) Y = cosx
![Page 2: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/2.jpg)
Test Review
• Retakes?
![Page 3: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/3.jpg)
Differentials
• We define the valuesas the difference between 2 values
These are known as differentials, and can also be written as dx and dy
![Page 4: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/4.jpg)
Linear Approximations
• The tangent line at a point of a function can be used to approximate complicated functions
• Note: The further away from the point of tangency, the worse the approximation
![Page 5: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/5.jpg)
Linear Approximation of df
• If we’re interested in the change of f(x) at 2 different points, we want
• If the change in x is small, we can use derivatives so that
![Page 6: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/6.jpg)
Steps
• 1) Identify the function f(x)• 2) Identify the values a and• 3) Use the linear approximation of
![Page 7: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/7.jpg)
Ex 1
• Use Linear Approximation to estimate
![Page 8: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/8.jpg)
Ex 2
• How much larger is the cube root of 8.1 than the cube root of 8?
![Page 9: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/9.jpg)
Ex 3,4
• In the book bc lots to type
![Page 10: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/10.jpg)
You try
• 1) Estimate the change in f(3.02) - f(3) if f(x) = x^3
• 2) Estimate using Linear Approximation
![Page 11: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/11.jpg)
Linearization
• Again, the tangent line is great for approximating near the point of tangency.
• Linearization is the method of using that tangent line to approximate a function
![Page 12: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/12.jpg)
Linearization• The general method of linearization1) Find the tangent line at x = a2) Solve for y or f(x) 3) If necessary, estimate the function by
plugging in for xThe linearization of f(x) at x = a is:
![Page 13: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/13.jpg)
Ex 1
• Compute the linearization ofat a = 1
![Page 14: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/14.jpg)
Ex 2
• Find the linearization of f(x) = sin x, at a = 0
![Page 15: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/15.jpg)
Ex 3
• Find the linear approximation to f(x) = cos x at and approximate cos(1)
![Page 16: 4.1 Linear Approximations Fri Oct 16 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx.](https://reader036.fdocuments.us/reader036/viewer/2022082911/5a4d1bb97f8b9ab0599d01c2/html5/thumbnails/16.jpg)
Closure
• Journal Entry: Use Linearization to estimate the square root of 37
• HW: p.214 #5 7 11 17 30 33 47 51 62 74