4.1 Linear Approximations Thurs Jan 7 Do Now Find the slope of each function at 1) Y = sinx 2) Y = cosx

Quiz Review Quiz retakes until Fri

Differentials We define the values as 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?

You try 1) Estimate the change in f(3.02) - f(3) if f(x) = x^3 2) Estimate using Linear Approximation

Closure Use Linear Approximate to estimate f(3.02) - f(3) if f(x) = x^4 HW: p.213 #1-13 odds, 17-21 odds

4.1 Linearization Fri Jan 8 Do Now Find the equation of the tangent line of at

HW Review p.213 #1-13 17-21 1) 0.12 19) -0.0005 3) -0.00222 21) 0.083333 5) 0.003333 7) 0.0074074 9) 0.04930 11) -0.03 13) -0.007 17) 0.1

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 linearization Find the tangent line at x = a Solve for y or f(x) If necessary, estimate the function by plugging in for x The linearization of f(x) at x = a is:

Ex 1 Compute the linearization of at 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)

Ex 4 Use linearization to approximate cos(1)

More examples Use a linear approximation to approximate

Closure Journal Entry: Use Linear Approximation to estimate the square root of 26 HW: p.214 #45-51 59-63 odds

Linear Approximation Practice Mon Jan 11 Do Now Use linear approximations to estimate

HW Review p.214 #45-51 59-63 45) L(x) = 4x - 3 47) L(x) = x - pi/4 + 1/2 49) L(x) = -1/2 x + 1 51) L(x) = 1 59) L(17) = 0.24219 61) L(10.03) = 0.00994 63) L(64.1) = 4.002083

Linearization Review We can use linear approximation (tangent line equations) for 2 uses: 1) Find the difference between to values of f(x) 2) Estimate the value of f(x) at specific points

Practice (green book) Worksheet p.249 #5-10, 17-22

Closure Hand in: Use linear approximation to estimate HW: Finish worksheet p.249 #5-10 17-22

HW Review p.249 #5-10 5) 6) 7) 8) 9) 10)

HW Review p.249 #17-22 17) .842 18) .788 19) 2.00125 20) 2.0025 21) 2.005 22) 1.030