we look at the function f : R? → R mit f(x, y) = y sin(2x)determain the tangential plane at point (6T, 4, f(67, 4)) and put it in the form vl 1 E: v2 + s + t 1 8, t eR v3 v4 v5 v1 v2 v3 v4 v5

Q: Define two vector functions F(t) = 8 sin(t)i + 6 cos(t)j + (t – 2)k ü(t) = 6 sin(t)i + 8cos(t)j + (t…

A: We have to find the dot product of r(t) and u(t)

Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: Find the directional derivative of the function at the given point in the direction of the vector v.…

A:

Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: B- Find the directional derivative of the function W = x² + xy + z³ at the point P: (2,1,1) in the…

A:

Q: c) Given the scalar function p(x, y, z) = x²e - y³ + cos 2z. Find the directional derivative of p(x,…

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Q: Define two vector functions 7(t) = 2 sin(t)i + 8 cos(t)j + t°k ü(t) = 8 sin(t)? + 2 cos(t)j + (t? –…

A:

Q: 10. r(t) = F(t) x G(t), where F(t) = t³i – tj + tk, G(t) G(t) = 3√/ti+ 1 t + īj + (t + 2) k -j 1

A:

Q: Let ß be the vector-valued function 3u B(u, v) = { 32 ß: (-2, 2) × (0, 2π) → R³, 3u² + 4 COS(V) } u…

A: Since you have asked multiple questions, we will answer the questions (a) and (b) for you. If you…

Q: One of the following vector functions is a gradient Select one: O (xy? + 2y?)7+ (2y³ – x²y+ 2x)j O…

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Q: (a) Find the direction for which the directional derivative of the function f(x, y, z) = √√x² + y² +…

A:

Q: Let f(x, y, z) = 5x³ - y³ + z². Find the maximum value M for the directional derivative at the point…

A: Our objective is given below:

Q: Q4:- Find the derivative of the function at Po in the direction of the vector AC. f(x, y, z) = x² +…

A:

Q: 6. Consider the vectors u, = 1 and u₂ 3, let W be the subspace spanned by u, and u₂. -2 a. Verify…

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Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

Q: determine wave direction in the function f(x,y) x^y — x²y³ decreases faster at the point (2,-3)

A: Wave direction in the function f(x,y)= x4y-x2y3 that decreases faster at the point (2,-3) is…

Q: (Q1)-A-State(First fundamental theorem) and Show that :(2₂0 /(10), +..) -B-Let f:( M, (Z).+-)-(M,…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: af Calculate at point (-1,2) where f (x, y) = 8+ x² + 3y? dv and v = i+j. 10 2 5/2 V2

A: fx, y=8+x2+3y2  &  v=i+j Partially differentiating with respect to x, we get fxx, y=0+2x+0=2x…

Q: 31. Ke [<cos (1₂) c² - 3+₁ ²+/=/=) +‡ -1 (-1,19,-27, t = 11. Ŕ(t) (1) show that I is continuous at t…

A: This is a problem of vector calculus.

Q: The temperature on a cubic box [0, 4] × [0, 4] × [0, 4] (measured in meters) can be described by the…

A: Function, Tx, y, z=x2y+y2z

Q: 4) Let f(x, y) = x² – 2xy a) Find the gradient of f at the point (1, –2). b) Use the result of (a)…

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Q: The directional derivative of f(x, y) = cos(x²y³ ) in the direction of the vecto v = 3i + 4j is Du…

A: The given function is: f(x,y)=cos(x2y3) So, ∇f(x,y)=∂f∂x,∂f∂y…

Q: Given the vector-valued function r (t) = e-t i +t j +(tan t) k Find: a) r'(t) b) r"(t) c) r'(t) .…

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Q: H is the set of all vectors x = X2 in R³ with the property that X3 X1 + x2 = 0 and Xi – x2 + x3 = 0.…

A:

Q: 2. Consider a function g(x, y, z) = 1 + ln(3x + 6y +9z) and a vector u = (0, -1, 1). (a) Find the…

A:

Q: Prove that the directional derivative of f(x, y) = y + 2xy³+x²y2 at point (0, 1) in the direction…

A: The gradient is just the vector of partial derivatives. The partial derivatives of f at the point…