lecture16 - Lecture 16 Lecture 16: (Sec.3.6) Implicit...

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Unformatted text preview: Lecture 16 Lecture 16: (Sec.3.6) Implicit Differentiation dy for the following: dx 1 1) y = 2x − 3 Find 2) xy + 2y − x = 0 But consider the following equation: 3) y − xy 2 + 2x = 1 1 Lecture 16 2 Recall the following form of the Chain Rule: If u is a differentiable function of y where y is a differentiable function of x, then Now find the following for y = f (x): d3 1) (y ) dy d3 2) (y ) dx d2 3) (x + y 3) dx 4) d2 (x y ) dx Lecture 16 3 dy for the equation dx 2 y − xy + 2x = 1. ex. Find To find 1) 2) dy by Implicit Differentiation: dx Lecture 16 4 dy given that x and y are related by the ex. Find dx equation x 2 y 3 + 2 − 3y = x 2 . Lecture 16 5 ex. Find the slope of the tangent line to the curve 1 x = x + 2y at (1, ). y 2 NOTE: Lecture 16 6 ex. Find the equation of the tangent line to the curve (x + y )2 − xy 2 = y + 3 at (−1, 2). Lecture 16 7 ex. Find the slope of √ tangent line to the graph the of x2 + y 2 = 9 at (2, − 5): 1) Explicitly Lecture 16 8 2) Find the slope of the tangent line to the graph of √ 2 2 x + y = 9 at (2, − 5) using implicit differentiation. Lecture 16 3) Show that the two forms of d2y 4) Find 2 if x2 + y 2 = 9. dx 9 dy are equivalent. dx Lecture 16 10 ex. Suppose that x and y are differentiable functions of t so that x2 + y 2 = 9. Find dx dy in terms of x, y, and . dt dt Lecture 16 11 dx if x2 + y 2 = 9, and y is decreasing at ex. Find dt a constant rate of 0.5 units/second when x = 1 and y > 0. Lecture 16 12 ex. Suppose a point is moving along the graph of xy 2 + 2y = −4 so that its x-coordinate is increasing at the rate of 1.5 inches per minute. How fast is the y -coordinate changing when y = −1? ...
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lecture16 - Lecture 16 Lecture 16: (Sec.3.6) Implicit...

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