Solve whole parts correctly in 30 minutes and get the thumbs up please show neat and clean work 2. Let f(x,y) = + (y + 2)² +2a) Sketch the level curve for f(x,y) = 6. (Ch. 10.5, 12.1)b) Find the gradient of f(x,y) in the point (4, -2). (Ch. 12.7)c) Let g(x,y) = ln(f(x,y)). In which points (x,y) in R2 is the function g welldefined. (Ch.10.5, 12.1)

Given function is fx,y=x24+(y+2)2+2 Solution a) Level curve for fx,y=6 is an ellipse x242+y+224=1

Answered: 2. Let f(x,y) = + (y + 2)² + 2 a)…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 41E

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17. Sketch the ioiicwing curve by cing seconá derivative: 1) y= 1+x 2) y=-x(x-7) 3) y (x+ 2) (x-3) 4) y=x(5-x) (ans.: max.(1,0.5); min.(-1,-0.5)) (ans.: max.(7,0); min.(2.3,-50.8)) (ans.: max.(-2,0); min.(1.3,-18.5)) (uns. mux.(3.5,18.5);ra0,0)) 18. What is the smallest perimeter possible for a rectangle of area 16 in.2 ? (ans.: 16) 19. Find the area of the largest rectangle with lower base on the x- axis and upper vertices on the parabola y 12-x. (ans.:32) 20) A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m. of fence at your disposal. What is the largest area you can enclose ? (ans.:80000) 21) Show that the rectangle that has maximum area for a given perimeter is a square. 22) A wire of length L is available for makıng a circie and a square. How should the wire be divided between the two shapes to maximize the sum of the enclosed areas? (ans.: all bent into a circle) 23) A closed container is made from a…
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a) Find the directional derivative of the function f(x, y, z) = x² y? (2z+1)² at the point P:(1, – 1, 1) and in the direction of a =[3, 3, 0]. b) Find the direction of maximum decrease of f at the point P. Maximum file size: 100MB. maximum number of files: 1
Draw the contour curves for h (x, y) _y²+x² -1+2x and c = 0, 2, 4.
1. A function of two variables is given f(x, y) = x² + y² + e-(x²+y²) a) Draw a picture of the graph of the function f. Use for example Geogebra. b) Find a tangent plane to the graph of the function f at all points (x, y, f(x, y)) c) At what points is the tangent plane parallel to the xy-plane? How does the function behave near the points (0, 0)?
3. Find the gradient Vf and the directional derivative at the point P (1,-1,2) in the direction a = (2,-1,1) for the function f(x,y,z) = x°z-yx² + z² . In which direction is the directional derivative at P decreasing most rapidly and what is its value?
Which of the following straight lines are tangent to the curve y = x³ + 4x2 + 7x and passes through the point (0 , 0)? (1) y = 7x (II) y = - 11x %3D (III) y = 11x (IV) 3x + y = 0 (V) 3x - y = 0 Lines I, and V O Lines I , and II Lines II , and IV O Lines III , and IV Only Line I
3. Let f(x) = 2x - x². (a) Find the secant line to the graph of f passing through the points (0,0) and (1,1). 1 (b) Find the tangent line to the graph of f at x = 2 (c) Sketch the graphs of f(x), the secant line, and tangent line on the same set of axes. -4 -3 -1 Y 5 2 4 -2 -4 6 2 3 A 5 x
- 3) The curve C has equation y = x³ - 11x + 1 a) Find the gradient at the point on C, where x = 3. The point P lies on C and the gradient at that point is 1. b) Find the possible coordinates of P. (Criteria 1.3)
5. C nts) Let F =, use Green's Theorem to evaluate F. dr where the curve C' is the curve shown in the picture consisting of functions y=x² − 2x going from point (0,0) to the point (3,3) followed by the line segment from (3,3) to (0,0). (0,0) y=x (3,3) C y=x²-2x

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2. Let f(x,y) = + (y + 2)² + 2 a) Sketch the level curve for f(x,y) = 6. (Ch. 10.5, 12.1) b) Find the gradient of f(x,y) in the point (4,-2). (Ch. 12.7) c) Let g(x,y) = ln(f(x,y)). In which points (x,y) in R2 is the function g welldefined. (Ch. 10.5, 12.1)