√√3 1 5.(2.6) (1) Let f(x, y) = ln(x² + y³) and v = derivative of f in the direction of v at x0 = Find Vf(x, y) and the directional = (2, 1). (0, 1). (2) Consider a curve given by x2 − 2xy + y³ = 1. Find a normal vector to the curve at x0 = 2
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- Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y) = 7ex sin(y), (0, ?/3), v = (-3,4)f(x, 3) In(x2 + Yof function P(3, 4) at the point of = (9, 12) Directional derivative in the direction of the vector to you.5. (a) Find the directional derivative of f(x, y) = √√√x² - y² at P(5,3) in the direction of v = (1, -1). (b) Find the unit vector u in the direction of which ƒ has the maximum rate of change at (5,3). What is this maximum rate of change?
- Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) =8xy2, (5,-7) maximum rate of change direction vector× Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = y²/x, (2, 4) direction of maximum rate of change (in unit vector) = < maximum rate of change =(1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =
- Let u(t) = 9t'i+ (² -1)j-6k and v(t)= e'i+5 ej- e k Compute the derivative of the following function. u(t) - v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the vector-valued function i+ k. O B. The derivative is the scalar function Click to select and enter your answer(s) and then click Check Answer. All parts showing Clear All OK earchFind the directional derivative of the function at the given point in the direction of the vector v. f(x, y) = 3e* sin(y), (0, π/3), v= (-3, 4) Dvf(0, π/3) = (-9√3+6) 2 X3. Consider the function f(z, y) = r'y - 4y at the point P(2, -1). (a) Find the dlirectional derivative of the function at the point P in the direction of the vector = (2,5). (b) In what direction does f have the maximum rate of change? What is the maxi- mum rate of change?
- Find the directional derivative of the function f(x, y) = ln (x³ + y) at the point (-2, 2) in the direction of the vector4-(12+13 p.) (a) Find the derivative of f(r, y, 2) = xe+ In(z2) at Po.0,) in the direction of 21 +j- 2k. (b) Find the equations for the tangent plane and the normal line at Po(1,-1,3) on the surface r + 2ry - y? + 2 = 7.Calculate the directional derivative of f(x, y) = x³y² in the direction of v = 2i - 2j at the point P = (2, 1). Remember to normalize the direction vector. Duf(2, 1) =