In Exercises 9-18, evaluate
C: Smooth curve from (0, 0) to (3, 8).
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Calculus: Early Transcendental Functions
- Let the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the integral ∫c F· dr = ∫c (2x-y)dx + (4y-x)dy in two different ways:(a) Parameterize the curve C and compute the integral directly. (b) Use the Fundamental Theorem of Line Integrals.arrow_forward2) r(1) = ti -t j-t'k, t20 Draw the graph of the vector-valued function, explaining it in detail.arrow_forwardDoes f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?arrow_forward
- Complexarrow_forward3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.arrow_forwardCalculate the derivative using Part 2 of the Fundamental Theorem of Calculus. & L² (6t²-t) 22 dt #1₁ (6²2-1) 22 dz 囡囡arrow_forward
- Evaluate F. dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. SF I (4z + 2y) dx + (2x − 3z) dy + (4x - 3y) dz (a) C: line segment from (0, 0, 0) to (1, 1, 1) (b) C: line segment from (0, 0, 0) to (0, 0, 1) to (1, 1, 1) (c) C: line segment from (0, 0, 0) to (1, 0, 0) to (1, 1, 0) to (1, 1, 1)arrow_forwardVerify directly that the real and imaginary parts of the following analytic functions satisfy Laplace’s equation: (a) f(z) = z 2 + 2z + 1 (b) g(z) = 1 z (c) h(z) = e zarrow_forwardUse Cauchy-Riemann equations to find all points z such that f is differentiable: (a) f(z)= (b) f(z) =|z|+ izarrow_forward
- I need help with this problem and an explanation for the solution described below (Vector-Valued Functions, Derivatives and integrals, Vector fields)arrow_forwardSuppose F(x, y) = x² + y² and C is the line segment from point A = (1, −1) to B = (3, −5). (a) Find a vector parametric equation (t) for the line segment C so that points A and B correspond to t= 0 and t = 1, respectively. F(t) = = (b) Using the parametrization in part (a), the line integral of along C' is dt LE · dF = [ ° F (F(t)) - 7"' (t) dt = √° with limits of integration a = and b = (c) Evaluate the line integral in part (b).arrow_forwardDetermine the Derivatives of Transformarrow_forward
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