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(45). (Symmetric Matrices and Quadratic Forms) Examine the following functions for relative extremum values: (a) F...
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e....
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
Find the extreme values of the following functions.(Write together the problem-solving process.) (1) f(x, y) =...
Find the extreme values of the following functions.(Write together the problem-solving process.) (1) f(x, y) = x^2 + 2xy + 2y^2 (2) f(x, y) = x^2 + 2xy − 2y^2 (3) f(x, y) = 3x^2 + 6xy − 2y^3
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4 9. Find the relative and absolute max and min of th...
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4
Problem 2. Write the Taylor series expansion for the following functions up to quadratic terms a....
Problem 2. Write the Taylor series expansion for the following functions up to quadratic terms a. cos (x) about the point x*-pi/4. b. f(xx) 10x1 20x^x2 +10x3 +x 2x 5 about the point (1,1). Compare the approximate and exact values of the function at the point (1.2, 0.8)
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form...
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form b) Write the vertex form (also known as standard form). c) Find the y-intercept. Which of the three forms listed above is the most useful in identifying the y-intercept and why? d) Find the x-intercepts. Which of the three forms listed above is the most useful in identifying the x-intercepts and why? e) Find the vertex. Which of the three forms listed above is...
5. Find the derivative matrices of the following vector-valued functions of several variables (a) x(t) (t cos t,t sin t...
5. Find the derivative matrices of the following vector-valued functions of several variables (a) x(t) (t cos t,t sin t) (is there one row or one column?) (b) f(r, y) (2r 4, x - 2y7, y 3) (c) f(x,y, 2)(xz2, x + zy?)
5. Find the derivative matrices of the following vector-valued functions of several variables (a) x(t) (t cos t,t sin t) (is there one row or one column?) (b) f(r, y) (2r 4, x - 2y7, y 3)...
(1) (2) (3) 5. (20 points) a) Solve the following system of equation for symbolic 2,...
(1) (2) (3) 5. (20 points) a) Solve the following system of equation for symbolic 2, 4, 2 3x - 2y +52 = 12 2-32 + y = -1 2-y-* = 4 b) State the code for solving the following ordinary differential equation 204 - 3y = t?, y(0) = 0,//(0) = 1. ata c) Plot the following symbolic functions a) f(x) = for symbolic r in (-2,2) interval. b) f(x,y) = sin(x2+x2) in (-5,5). - 2 at (4)
10. Use the limit definition of the derivative to calculate the derivatives of the following functions....
10. Use the limit definition of the derivative to calculate the derivatives of the following functions. a. f(x) = 2x2 – 3x + 4 b. g(x) = = x2 +1 1 x2 +1 c. h(x) = 3x - 2 a. 11. Find the derivative with respect to x. x² - 4x f(x)= b. y = sec v c. 5x2 – 2xy + 7y2 = 0 1+cos x 1-cosx cos(Inu) e. S(x) = du 1+1 + + f. y =sin(x+y) g....
1. Find the maxima of the following functions. (a) f(x)-2-4. )2 (c) f (z,y)2+3. (d) f...
1. Find the maxima of the following functions. (a) f(x)-2-4. )2 (c) f (z,y)2+3. (d) f (x,y) = xy-x2-y2 + 9y.
Questions A and E (b) f(x, y) = 2.2-2y2-6x + 8y + 3 (c) f(z, y)=x2...
Questions A and E
(b) f(x, y) = 2.2-2y2-6x + 8y + 3 (c) f(z, y)=x2 + 6ry + 2/-6x +10y-2 Constrained Optimization 3. Find the values of r and y that maximize (or minimize) f(x,y) subject to the following constraints (a) f(z, y) = ry, subject to x + 2y = 2 (b) f(z, y)y+4), subject to z+y 8 (c) f(x,y) = x-3y-xy, subject to x + y = 6 (d) f(x, y)-7-y 2, subject to z ty 0...
XTAX=1. determine their canon- 1. Write the following quadratic forms as V(x) ical forms, find the modal matrices (i.e. the matrices of unit eigenvectors) of the corresponding transformations and write down explicite expressions for canonical cOordinates (y1, 2, y3) in terms of the original coordinates (x1, X2, X3). State what surfaces these quadratic forms correspond to = > (a) x + 4x1r2 + 4a13-8a2x3 = 1; (b) a3a3a^ + 4xj2 +4x131223 1; (c) 4a7 2a2 2axjx2 2x13+ 6x23 = 1....
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4
Problem 2. Write the Taylor series expansion for the following functions up to quadratic terms a. cos (x) about the point x*-pi/4. b. f(xx) 10x1 20x^x2 +10x3 +x 2x 5 about the point (1,1). Compare the approximate and exact values of the function at the point (1.2, 0.8)
13. The following quadratic function is written in general form: f(x)=x2-6x-7 a) Write the factored form b) Write the vertex form (also known as standard form). c) Find the y-intercept. Which of the three forms listed above is the most useful in identifying the y-intercept and why? d) Find the x-intercepts. Which of the three forms listed above is the most useful in identifying the x-intercepts and why? e) Find the vertex. Which of the three forms listed above is...
5. Find the derivative matrices of the following vector-valued functions of several variables (a) x(t) (t cos t,t sin t) (is there one row or one column?) (b) f(r, y) (2r 4, x - 2y7, y 3) (c) f(x,y, 2)(xz2, x + zy?)
5. Find the derivative matrices of the following vector-valued functions of several variables (a) x(t) (t cos t,t sin t) (is there one row or one column?) (b) f(r, y) (2r 4, x - 2y7, y 3)...
(1) (2) (3) 5. (20 points) a) Solve the following system of equation for symbolic 2, 4, 2 3x - 2y +52 = 12 2-32 + y = -1 2-y-* = 4 b) State the code for solving the following ordinary differential equation 204 - 3y = t?, y(0) = 0,//(0) = 1. ata c) Plot the following symbolic functions a) f(x) = for symbolic r in (-2,2) interval. b) f(x,y) = sin(x2+x2) in (-5,5). - 2 at (4)
10. Use the limit definition of the derivative to calculate the derivatives of the following functions. a. f(x) = 2x2 – 3x + 4 b. g(x) = = x2 +1 1 x2 +1 c. h(x) = 3x - 2 a. 11. Find the derivative with respect to x. x² - 4x f(x)= b. y = sec v c. 5x2 – 2xy + 7y2 = 0 1+cos x 1-cosx cos(Inu) e. S(x) = du 1+1 + + f. y =sin(x+y) g....
1. Find the maxima of the following functions. (a) f(x)-2-4. )2 (c) f (z,y)2+3. (d) f (x,y) = xy-x2-y2 + 9y.
Questions A and E
(b) f(x, y) = 2.2-2y2-6x + 8y + 3 (c) f(z, y)=x2 + 6ry + 2/-6x +10y-2 Constrained Optimization 3. Find the values of r and y that maximize (or minimize) f(x,y) subject to the following constraints (a) f(z, y) = ry, subject to x + 2y = 2 (b) f(z, y)y+4), subject to z+y 8 (c) f(x,y) = x-3y-xy, subject to x + y = 6 (d) f(x, y)-7-y 2, subject to z ty 0...
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