Question

Calculate y^(k)(0) for 0 ≤ k ≤ 5, where y = 4x⁴ + ax³ + bx² + cx + d (with a, b, c, d being constants).

y^(0)(0) =
y^(1)(0) =
y^(2)(0) =
y^(3)(0) =
y^(4)(0) =
y^(5)(0) =

Answer 1

To calculate y^(k)(0) for 0 ≤ k ≤ 5, evaluate the kth derivative of the function at x=0. The values are: y^(0)(0) = d, y^(1)(0) = c, y^(2)(0) = 2b, y^(3)(0) = 6a, y^(4)(0) = 96, y^(5)(0) = 0.

Step-by-step explanation:

To calculate y^(k)(0) for 0 ≤ k ≤ 5, we need to find the kth derivative of the function y = 4x⁴ + ax³ + bx² + cx + d, and then evaluate it at x=0. Let's find the derivatives of the function:

y^(0)(x) = 4x⁴ + ax³ + bx² + cx + d

y^(1)(x) = 16x³ + 3ax² + 2bx + c

y^(2)(x) = 48x² + 6ax + 2b

y^(3)(x) = 96x + 6a

y^(4)(x) = 96

y^(5)(x) = 0

Now, let's substitute x=0 into each derivative:

y^(0)(0) = d

y^(1)(0) = c

y^(2)(0) = 2b

y^(3)(0) = 6a

y^(4)(0) = 96

y^(5)(0) = 0