Given that Z = x2 + y2 + x2y', Evaluate (i) dz/dx (ii) dz/dy, when x = 1 and y = 1. If y = ae^x + be2x, show…

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Given that Z = x2 + y2 + x2y', Evaluate (i) dz/dx (ii) dz/dy, when x = 1 and y = 1. If y = ae^x + be2x, show that y'' - 3y' + 2y = 0. A body moves in a straight line according to the law of motion s = t3 - 2t2 + t. Find velocity and acceleration after 3 seconds.

ScanSolve AI Tutor · Accepted Answer

here's the solution for Question 5a: (5a). Given that Z = x² + y² + x²y³, Evaluate (i) (∂ Z)/(∂ x) (ii) (∂ Z)/(∂ y), when x=1 and y=1 Step 1: Find the partial derivative of Z with respect to x, denoted as (∂ Z)/(∂ x). When differentiating with respect to x, treat y as a constant. Z = x² + y² + x²y³; (∂ Z)/(∂ x) = (∂)/(∂ x)(x²) + (∂)/(∂ x)(y²) + (∂)/(∂ x)(x²y³); (∂ Z)/(∂ x) = 2x + 0 + 2xy³; (∂ Z)/(∂ x) = 2x + 2xy³; Step 2: Evaluate (∂ Z)/(∂ x) at x=1 and y=1. Substitute x=1 and y=1 into the expression for (∂ Z)/(∂ x). (∂ Z)/(∂ x) |_(1,1) = 2(1) + 2(1)(1)³; (∂ Z)/(∂ x) |_(1,1) = 2 + 2(1)(1); (∂ Z)/(∂ x) |_(1,1) = 2 + 2; (∂ Z)/(∂ x) |_(1,1) = 4 Step 3: Find the partial derivative of Z with respect to y, denoted as (∂ Z)/(∂ y). When differentiating with respect to y, treat x as a constant. Z = x² + y² + x²y³; (∂ Z)/(∂ y) = (∂)/(∂ y)(x²) + (∂)/(∂ y)(y²) + (∂)/(∂ y)(x²y³); (∂ Z)/(∂ y) = 0 + 2y + x²(3y²); (∂ Z)/(∂ y) = 2y + 3x²y²; Step 4: Evaluate (∂ Z)/(∂ y) at x=1 and y=1. Substitute x=1 and y=1 into the expression for (∂ Z)/(∂ y). (∂ Z)/(∂ y) |_(1,1) = 2(1) + 3(1)²(1)²; (∂ Z)/(∂ y) |_(1,1) = 2 + 3(1)(1); (∂ Z)/(∂ y) |_(1,1) = 2 + 3; (∂ Z)/(∂ y) |_(1,1) = 5

Given that Z = x2 + y2 + x2y', Evaluate (i) dz/dx (ii) dz/dy, when x = 1 and y = 1. If y = ae^x + be2x, show that y'' - 3y' + 2y = 0. A body moves in a straight line according to the law of motion s = t3 − 2t2 + t. Find velocity and acceleration after 3 seconds.

Step-by-step solution

Find the partial derivative of $Z$ with respect to $x$ , denoted as $\frac{\partial Z}{\partial x}$ .

Evaluate $\frac{\partial Z}{\partial x}$ at $x=1$ and $y=1$ .

Find the partial derivative of $Z$ with respect to $y$ , denoted as $\frac{\partial Z}{\partial y}$ .

Evaluate $\frac{\partial Z}{\partial y}$ at $x=1$ and $y=1$ .

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More Mathematics Questions

Given that Z = x2 + y2 + x2y', Evaluate (i) dz/dx (ii) dz/dy, when x = 1 and y = 1. If y = ae^x + be2x, show that y'' - 3y' + 2y = 0. A body moves in a straight line according to the law of motion s = t3 − 2t2 + t. Find velocity and acceleration after 3 seconds.

Step-by-step solution

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