Calculus on Demand at Dartmouth College Lecture 13 | Index | Lecture 15
Lecture 14


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In this lecture we look at two applications of the derivative: finding the zeroes of a function, and approximating the value of a function with a line. The former application leads to a numerical procedure called "Newton's Method" which, for example, is a popular technique used by calculators to find square roots.

Quick Question

What is the equation of the line tangent to the graph of ln(x) at x = e?



Newton's Method
Linear Approximations

Today's Homework

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Newton's Method Quiz
Linear Approximations Quiz


  • Click to see the exampleConsider the equation x1/3 = 0 and say you intend to solve it using Newton's method. If the initial value chosen is not 0, what will happen when you apply Newton's method?
  • Click to see the exampleUsing Newton's Method, find the equation of a line tangent to the curve y = sin(x) that passes through the origin. Give the answer to 6 decimal places.
  • Click to see the exampleUse Newton's Method to approximate to 6 decimal places the solution to the equation tan(x/4) = 1.
  • Click to see the exampleWhen solving problems in geometric optics, engineers and physicists often use the simplifying assumption that, for small angles θ, sin(θ) is approximately equal to θ. Find a linear approximation for sin(x) that shows why this is a reasonable assumption.
  • Click to see the exampleA pizza restaurant sells an average of 80 pizzas per day at its usual price of $12.95. It experiments with sales and coupons for dollars off the usual price, and finds that the number of pizzas sold when the price decreases by 2 dollars is 135. It estimates that the number of pizzas sold when the price goes down by x dollars is modeled by the function 50 ln(x + 1) + 80. Use linear approximation to find the change in the number of pizzas sold when the price drops from $10.95 to $9.95.
  • Click to see the exampleFind the linear approximation of the function f(x) = √(1 − x) about a = 0. Use it to approximate the square root of 0.9.


  • Click to see the appletNewton's Method
  • Click to see the appletBest Linear Approximation


  • click to see the videoNewtons method
  • click to see the videoNewton's method might go wrong
  • click to see the videoNewton's method bites the dust
  • click to see the videoFind the equation for the tangent line of 4x2 + 10x + 5 at the point (2,1) using power rule to find formula for derivative
  • click to see the videocos(x) at π/3

Lecture 13 | Index | Lecture 15