Q1 (9 points) Let z(x, y) and w(x, y) be smooth functions that satisfy the equations w+w³ + x²yz + 4 = 0, z³+w+xy/w+1=0 and also satisfy 1 z = 1, w = -1 when x = -29 y 2 Using differentials, estimate the value of z(-1.95, -.35) and calculate the percent relative error (the actual value is z(-1.95, —.35) = 0.910558).

The directional derivative of o = 5x2y – 5y2z + 2.5z2x at the point P(1, 1, 1) in the direction of the line * X-1 2 y-3 = Z -2

Use method of variation of parameters. y" +2y-8y=2e-2x _ e - .

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…

Find the directions of maximum and minimum change of f at the given point, and the values of the maximum and minimum rates of change. f{x, y) = 4y²e*, (2, –4) The maximum change is 32e V37 ; in the direction (-192e“, 32e®); The minimum change is –32e V37 ; in the direction (192e", -32e). The maximum change is 32e6 The minimum change is –32e 6 V37 :; in the direction (192e, -32e ). V37 ; in the direction (-192e6, 32e ); The maximum change is 32e V37 ; in the direction (192e, -32e); The minimum change is –32e V37 ; in the direction (-192e", 32e). The maximum change is 32e V37 ; in the direction (192e ,–32e ); The minimum change is –32e 6 V37 :; in the direction (-192e, 32e ).

Calculate y" and y" y = x² (8 x + 3)

Evaluate y' at the point (2,2) 5x3y - 2x2 = 72

of and ду of Consider the function f(x,y) = 9x' - 4y - 2. Find -

//welro) Find a local linear approximation for the function 5х + 6 f(x) valid for x near 5 5x + 6 Answer: Hint: The local linear approximation at x = 5 is the tangent line at x = 5. The equation for this tangent line, in slope-intercept form, is у 3 т(х — 5) +b where m f' (5) and b = f(5). So compute m and b, put these into the equation for the tangent line, and then enter the right-hand side of that equation as your answer.

Find ∂y/∂x1 and ∂y/x2 y = 2x13 − 11x12x2 + 3x22 y = 7x1 + 6x1x22 - px23 y = (2x1 + 3)(x2 - 2) y = (5x1 + 3)(x2 - 2) y = (2x1 - 3x2) / (x1 + x2) y = (x12 - 1) / (x1x2)