Practice Problems #8 (7092662) Current Score: Question Points 1. 0/59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0/1 0/1 0/1 0/1 0/2 0/2 0/3 0/3 0/6 0/2 0/13 0/1 0/5 0/6 0/2 0/7 0/3 0/1 points Total 0/59 SCalcET7 14.2.037. [2004965] - SCalcET7 14.2.501.XP. [1889082] - SCalcET7 14.2.503.XP. [1889071] - SCalcET7 14.2.504.XP. [1898639] - Determine the set of points at which the function is continuous. {(x, y) | x } and y {(x, y) | (x, y) ≠ (0, 0)} {(x, y) | x and y ≠ 0} {(x, y) | x · y ≠ 0} {(x, y) | x > 0 and y > 0} 2. 0/1 points Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim y4 (x, y) → (0, 0) x4 + 2y4 3. 0/1 points Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim xy cos y (x, y) → (0, 0) 5x2 + y2 4. 0/1 points Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim 18x3y (x, y) → (0, 0) 5x4 + y4 5. 0/2 points SCalcET7 14.3.005. [1888968] - SCalcET7 14.3.017. [1898016] - Determine the signs of the partial derivatives for the function f whose graph is shown below. (a) fx(x0, y0) positive negative (b) fy(x0, y0) positive negative 6. 0/2 points Find the first partial derivatives of the function. f(x, t) = e−8t cos πx fx(x, t) = ft(x, t) = 7. 0/3 points SCalcET7 14.3.032. [1898020] - SCalcET7 14.3.033. [1898031] - SCalcET7 14.3.041.MI.SA. [1724196] - Find the first partial derivatives of the function. f(x, y, z) = 4x sin(y − z) fx(x, y, z) = fy(x, y, z) = fz(x, y, z) = 8. 0/3 points Find the first partial derivatives of the function. w = ln(x + 8y + 4z) ∂w = ∂x ∂w = ∂y ∂w = ∂z 9. 0/6 points This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Find the indicated partial derivatives. 10. 0/2 points SCalcET7 14.3.049. [1857237] Use implicit differentiation to find ∂z/∂x and ∂z/∂y. e3z = xyz ∂z = ∂x ∂z = ∂y - 11. 0/13 points SCalcET7 14.3.056.MI.SA. [1724053] - This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Find all the second partial derivatives. 12. 0/1 points SCalcET7 14.4.003. [1905048] - SCalcET7 14.4.015. [1853571] - Find an equation of the tangent plane to the given surface at the specified point. z= 13. xy , (2, 2, 2) 0/5 points Explain why the function is differentiable at the given point. f(x, y) = 9 + e−xycos y, (π, 0) The partial derivatives are fx(x, y) = fx(π, 0) = and fy(x, y) = and fy(π, 0) = , so . Both fx and fy are continuous functions, so f is differentiable at (π, 0). Find the linearization L(x, y) of f(x, y) at (π, 0). L(x, y) = 14. 0/6 points SCalcET7 14.4.018. [1905011] Verify the linear approximation at (2π, 0). f(x, y) = Left f(x, y) = y + cos2x ≈ 1 + 1 y 2 y + cos2x . Then fx(x, y) = continuous functions for y > fx(2π, 0) = and fy(2π, 0) = and fy(x, y) = . Both fx and fy are , so f is differentiable at (2π, 0) by this theorem. We have , so the linear approximation of f at (2π, 0) is f(x, y) ≈ f(2π, 0) + fx(2π, 0)(x − 2π) + fy(2π, 0)(y − 0) = . - 15. 0/2 points SCalcET7 14.5.016. [1905019] - Suppose f is a differentiable function of x and y, and g(r, s) = f(8r − s, s2 − 2r). Use the table of values below to calculate gr(1, 8) and gs(1, 8). f g fx fy (0, 62) 1 4 5 2 (1, 8) 4 1 8 7 gr(1, 8) = gs(1, 8) = 16. 0/7 points SCalcET7 14.5.035.MI.SA. [1724042] - This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by the following function, where x and y are measured in centimeters. The temperature function satisfies Tx(5, 10) = 9 and Ty(5, 10) = 8. How fast is the temperature rising on the bug's path after 2 seconds? 17. 0/3 points SCalcET7 14.5.501.XP. [1905050] Use the Chain Rule to find the indicated partial derivatives. z = x4 + xy3, ∂z , ∂z , ∂z ∂u ∂v ∂w ∂z = ∂u ∂z = ∂v ∂z = ∂w Assignment Details x = uv4 + w3, y = u + vew when u = 1, v = 1, w = 0 -
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