Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Implicit differentiation Sometimes we are given an equation that implicitly defines a relationship between x and y. For example, we could have: x2 + y 2 = 1. dy We’d still like to find dx . Differentiate both sides of the equation with respect to x, treating y as if it were a function of x. [Use the Chain Rule, since y is still a function of x.] 1 For example, above we had x2 + y 2 = 1. Differentiating both sides with respect to x, we get: 2x + 2y dy = 0, dx which becomes dy −2x x = = . dx 2y y 2 Examples: dy in the following situations: Find dx 3x − 5y + 1 = 0 x2 + xy − 5y = 0 x2y − 3xy = x y 3 − 3x2 + 5x − 1 = xy 3 Example Take an ellipse given by 3x2 + 4y 2 = 16. Find the equation of the tangent line to this curve at the point (2, 1). 4 Example Suppose that a producer as two sources of raw iron, x and y, and that for given values (in tons) the amount of steel that can be produced is x2 + 3xy + y 2 . S= 10 Currently, the producer gets 10 tons of iron x and 20 tons of iron y. Estimate how much more of iron y will be needed if only 9 tons of iron x are available, in order to keep production of steel constant. 5 Related rates In a related rates problem, we typically have three variables. We have time t, and then two functions x and y which both depend on t. We have an equation that x and y satisfy, and we want to dy relate dx and dt dt . We use implicit differentiation again. 6 Example Suppose that the area of a circle is growing at a rate of 1.2 cm/s. How fast is the radius growing when the radius is 5cm? 7 Example A ladder 10 feet long is resting against a wall. If the bottom of the ladder slips, and slides away from the wall at a rate of 1 foot per second, how fast is the top of the ladder moving down the wall when the bottom of the ladder is 8 feet from the wall? 8