Unit 5 : Day 6 Linear Approximations,

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Unit 5 : Day 6 Linear Approximations, Inverse Functions, and the Mean Value Theorem Greg Kelly, Hanford High School, Richland, Washington

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the tangent the linearization of the function.

Linearization is just a tangent line with different notation: linearization of f at a is the standard linear approximation of f at a. The linearization is the equation of the tangent line, and you can use the old formulas if you like.

Linearization Example: Find the linearization L(x) of f(x) at x = a. How accurate is the approximation L(a + 0.1) ≈ f(a + 0.1)? The linearization is 0.0002 above the function.

Important linearizations for x near zero:

Inverse Functions and Derivatives At x = 2: We can find the inverse function as follows: To find the derivative of the inverse function: Switch x and y.

Slopes are reciprocals. At x = 2: At x = 4:

Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: The derivative of Derivative Formula for Inverses: evaluated at is equal to the reciprocal of the derivative of evaluated at .

A typical problem using this formula might look like this: Given: Find: Derivative Formula for Inverses:

a. Find the linearization of at x = 0. Practice Problems a. Find the linearization of at x = 0. b. How accurate is L(a + 0.1) compared to f(a + 0.1)? L(0.1)=1.05 f(0.1) = 1.049 The linearization is 0.001 above f(x). 2. Given f(2) = 5 and , find

If you travel from here to the beach (about 130 miles away) and you get there in 2 hours, what was your average speed? Mean Value Theorem for Derivatives Rate = Does that mean that at some time your actual speed had to be 65 mph? YES!!! The Mean Value Theorem says that at some point in the closed interval, the actual (instantaneous) slope equals the average slope. The Mean Value Theorem only applies over a closed interval.

If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b: Mean Value Theorem for Derivatives The Mean Value Theorem – remember in math words this means instantaneous slope = average slope. It is your job to find the instant (the c) that this happens. The Mean Value Theorem only applies over a closed interval.

Tangent parallel to secant. Slope of tangent: Slope of secant:

Online Practice for MVT Mean Value Theorem Click on Link above