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Unit 5: Applications of Differentiation I — Rates, Approximation, and the Mean Value Theorem

Two quantities linked through a shared equation, a tangent line standing in for a curve, and the theorem guaranteeing a derivative matches an average rate.

8 topics · 31 questions

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4.2 Linear Approximation and the Tangent Line

  1. How it worksHow does the tangent line at a known point let you estimate a function's value nearby without a calculator?
  2. Common mistakeWhy isn't a linear approximation exactly equal to the function's true value away from the point of tangency?
  3. Worked exampleFind a linear approximation for (8.1)1/3 using the tangent line to f(x) = x1/3 at x = 8.
  4. Worked exampleFind a linear approximation for the square root of 4.1 using the tangent line to f(x) = square root of x at x = 4.
Your turn1 of 2
Find the linearization L(x) of f(x) = x1/3 at x = 6.

4.2 Differentials and Estimating Error

  1. How it worksHow do dx and dy stand in for a small nudge in x and the tangent line's matching nudge in y?
  2. Worked exampleFind the differential estimate of the change in volume of a sphere when its radius grows from 5 cm to 5.1 cm.
  3. Common mistakeWhy isn't dy the same thing as the actual change in y, delta y, for a curved function?
Your turn1 of 2
Use differentials to estimate the change in volume dV of a sphere, as a function of its radius r, given a small radius increase of 0.224 cm.

5.1 Rolle's Theorem

  1. How it worksProve Rolle's Theorem using the Extreme Value Theorem and a case check on where the maximum or minimum occurs.
  2. Worked exampleFind the point guaranteed by Rolle's Theorem for f(x) = x2 - 4x + 3 on the interval from 1 to 3.
  3. Worked exampleShow that f(x) = sin(x) satisfies the hypotheses of Rolle's Theorem on the interval from 0 to π and find the guaranteed point.
  4. Common mistakeWhy doesn't Rolle's Theorem apply to a function that is not differentiable everywhere on the open interval?
  5. Going furtherWhat happens to Rolle's Theorem's guarantee when the function's two endpoint values are close but not exactly equal?
Your turn1 of 2
Find the point guaranteed by Rolle's Theorem for f(x) = (x - 3)(x - 8) on the interval from 3 to 8.

4.4 The Mean Value Theorem, Proved

  1. How it worksHow does building a helper function out of f(x) and the secant line reduce the Mean Value Theorem to Rolle's Theorem?
  2. Worked exampleProve the Mean Value Theorem by applying Rolle's Theorem to g(x) = f(x) minus the secant line through the endpoints.
  3. Common mistakeWhy isn't the point guaranteed by the Mean Value Theorem always the midpoint of the interval?
  4. Going furtherWhat happens to the Mean Value Theorem's guaranteed point as the interval's two endpoints move closer together?
  5. Worked exampleFind the value guaranteed by the Mean Value Theorem for f(x) = x3 on the interval from 0 to 2.
Your turn1 of 2
Find the value guaranteed by the Mean Value Theorem for f(x) = 4x3 on the interval from 0 to 6.

4.4 What the Mean Value Theorem Guarantees About Motion

  1. How it worksHow does the Mean Value Theorem prove that a function with zero derivative everywhere has to be constant?
  2. Worked exampleShow that two functions with the same derivative everywhere on an interval can only differ by a constant.
  3. Going furtherWhat happens to the Mean Value Theorem's guarantee when the function fails to be continuous at just one endpoint?
Your turn1 of 2
A car travels 309 miles in 6 hours. By the Mean Value Theorem, find the average speed the car must have matched at some instant during the trip.