Example 3.4 Find the directional derivative of g(x, y) = ex2cos y at (1, π)in the direction −3i + 4j.
Question
Example 3.4 Find the directional derivative of g(x, y) = ex2cos y at (1, π)in the direction −3i + 4j.
Solution
Sure, here are the steps to find the directional derivative:
Step 1: Compute the gradient of g(x, y) = e^(x^2)cos(y)
The gradient of a function is a vector that contains the first derivative of the function with respect to each variable. In this case, we have two variables, x and y.
The partial derivative of g with respect to x is: dg/dx = 2x * e^(x^2) * cos(y)
The partial derivative of g with respect to y is: dg/dy = -e^(x^2) * sin(y)
So, the gradient of g is ∇g = (2x * e^(x^2) * cos(y))i - (e^(x^2) * sin(y))j
Step 2: Evaluate the gradient at the point (1, π)
∇g(1, π) = (2 * e * cos(π))i - (e * sin(π))j = -2ei - 0j = -2ei
Step 3: Normalize the direction vector -3i + 4j
The direction vector is v = -3i + 4j. The magnitude of v is ||v|| = sqrt((-3)^2 + 4^2) = 5
The normalized direction vector is v_hat = v / ||v|| = (-3/5)i + (4/5)j
Step 4: Compute the directional derivative
The directional derivative of g at (1, π) in the direction v is given by the dot product of the gradient of g at (1, π) and the normalized direction vector v_hat.
D_v g(1, π) = ∇g(1, π) . v_hat = -2e * (-3/5) + 0 * (4/5) = 6e/5
So, the directional derivative of g(x, y) = e^(x^2)cos(y) at (1, π) in the direction -3i + 4j is 6e/5.
Similar Questions
Letf : R^2 -> R: (x,y) -> x^4 + y^2 + 2x^2y -1Answer the following:(a) Graph f(x, 0) against x in two dimensions and graph f(0, y) against y in two dimensions.(b) Draw the countour set f(x, y) = 3(c) Calculate the directional derivative of f at the point (1, 1) in the direction of thevector (−1, 1).(d) At the point (1, 1), find the direction of steepest ascent. That is, find the vector v∗ with length 1 that causes f to increase most rapidly.(e) Find the equation of the tangent plane at the point (1, 1) and comment on the relationship between the tangent plane and the slope of steepest ascent.
directional derivative
Find the directional derivative of f=xyz at (1,1,1) in the direction of 𝑖⃗ +𝑗⃗ +𝑘⃗
Let f (x, y, z) = sin(xy + z), and P be the point (0, −2, π3 ).(a) (6 pts) Compute ∇f (x, y, z).(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.(c) (4 pts) Calculate the directional derivative ∂f∂u (P ), where u is a unit vector making an angleπ6 with the gradient ∇f (P ).(d) (3 pts) The level surface f (x, y, z) = √32 defines z implicitly as a function of x and y near P .Compute ∂z∂x at P .
Find the directional derivative of V · (V f) at the point ( 1,-2, l) in the direction of normal to thesurface xy 2 z = 3x + z 2 , where f(x, y, z) = 2x 3 y2 z 4
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.