Post on 19-Jan-2018
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4.1 Linear ApproximationsFri Oct 16
Do NowFind the equation of the tangent
line of each function at 1) Y = sinx
2) Y = cosx
Test Review
• Retakes?
Differentials
• We define the valuesas the difference between 2 values
These are known as differentials, and can also be written as dx and dy
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
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
Steps
• 1) Identify the function f(x)• 2) Identify the values a and• 3) Use the linear approximation of
Ex 1
• Use Linear Approximation to estimate
Ex 2
• How much larger is the cube root of 8.1 than the cube root of 8?
Ex 3,4
• In the book bc lots to type
You try
• 1) Estimate the change in f(3.02) - f(3) if f(x) = x^3
• 2) Estimate using Linear Approximation
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
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:
Ex 1
• Compute the linearization ofat a = 1
Ex 2
• Find the linearization of f(x) = sin x, at a = 0
Ex 3
• Find the linear approximation to f(x) = cos x at and approximate cos(1)
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