L24 Math 233 Lecture Notes - Lecture 17: Unit Vector, Directional Derivative, Partial Derivative

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L24 Math 233 Lecture Notes - Lecture 17: Unit Vector, Directional Derivative, Partial Derivative

L24 Math 233 Lecture Notes - Lecture 18: Level Set, Directional Derivative, Implicit Function

L24 Math 233 Lecture Notes - Lecture 19: Level Set, Nissan L Engine, Unit Vector

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L24 math 233 - lecture 17 - chain rule and dependency. Explanation of the chain rule for a function f of two variables composed of function r with one variable: Function of two variables goes from two dimensions to one dimension- 2 . Function r goes from one dimension to two dimensions- 2. To compute the derivative of f composed with r: (t) Find the missing components that make up the equation: )(t) and ( d dt d dt (f. X"(t) = 2 and y"(t) = 3t^2 = 3*1^2 = 3 plug in d dt. Simplifies to (x, ) y = f. Note: hypothesis that f is differentiable insures that d f(x, ) 3 * ( x x f and c. + * y (1)) (1)) (y dx y * y0 y * ( x0 + y (x0, y0) y f d. R (f dt will be sufficiently accurate to (x, ) f equal.

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Calculus 3 help. This is a take home, open book, quiz, but Ineed help on it! I am mainly stuck on the last 4. If I could justget steps on how to approach the answers, that would help me a lot!Thank you in advance!

1. (45 marks) Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answer. Unless otherwise stated, it is not necessary to simplify your answers in this question. a) Find the equation of the tangent plane to the graph of the function z = f(x,y) = sell at (0,1). Answer b) Evaluate (T, 0) if f(x,y) = sin(ry). Answer c) Graph of a function z = f(x,y) is given by: Which of the following level curves correspondes to this function. Answer d) If f(x,y) = e(3x – 2y), find lim - Math)-f(x,y) e) Find % if f(x, y) = x2 - y3 and 2 = r sin(t), y = r cos(t). Answer f) Evaluate the directional derivative of f(x, y) = !at (1, -1) in the direction of u =(2, 1). Answer 8) Does Simono converge? h) Sketch the direction field of the differential equation = 1-2y. i) Consider the differential equation ay = 15 - 31, y(0) = 0, and use Euler's method with step size 0.1 to estimate y(0.1). Answer j) Knowing that (-3,-3) is a critical point of f(x,y) = 2:3 + y2 – 4.cy -3.0, use the second derivative test to find out whether it is a local minimum, local maximum, or a saddle point. Answer k) Solve the initial value problem, dy dt 642 2y + cos(u) y(1) = . Answer 1) Is ū=(2,2, -1) perpendicular to v = (5,-4, 2)? Answer m) Find the area between the graph of y = 1 + sin(x) and y=1-sin(x) from 0 to 7. Answer n) Determine the lentgh of the curve y = 8/2,0 SIS 3. Evaluate and simplify your answer. Answer o) Express the volume of the solid obtained by rotating the bounded region that lies between curves y = (x-2)2 and y=1,150 $4 about the vertical line 2 = 5. Do not evaluate this integral. Answer

L24 Math 233 Lecture Notes - Lecture 17: Unit Vector, Directional Derivative, Partial Derivative

L24 Math 233 Lecture Notes - Lecture 17: Unit Vector, Directional Derivative, Partial Derivative

L24 Math 233 Lecture Notes - Lecture 18: Level Set, Directional Derivative, Implicit Function

L24 Math 233 Lecture Notes - Lecture 19: Level Set, Nissan L Engine, Unit Vector

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Calculus

Single Variable Calculus: Early Transcendentals

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