Explanation

To find the slope of the tangent to the curve y³x + y²x² = 6 at (2, 1), we'll use implicit differentiation:

Given: y³x + y²x² = 6

Differentiating both sides with respect to x:

y³ + x_3y²(dy/dx) + y²_2x + x²*2y(dy/dx) = 0

Rearranging terms to isolate dy/dx:

(3xy² + 2x²y)(dy/dx) = -y³ - 2xy²

dy/dx = (-y³ - 2xy²) / (3xy² + 2x²y)

Now, evaluating dy/dx at (2, 1):

dy/dx = (-(1)³ - 2_2_(1)²) / (3_2_(1)² + 2*(2)²*1)

= (-1 - 4) / (6 + 8)

= -5 / 14

Explanation

Given that 1 - i is a root of the equation x² + ax + b = 0, and the coefficients are real, the complex conjugate 1 + i must also be a root.

The sum of the roots = (1 - i) + (1 + i) = 2 = -a

So, a = -2

The product of the roots = (1 - i)(1 + i) = 1² - i² = 1 + 1 = 2 = b

So, b = 2

Now, a - b = -2 - 2 = -4

Explanation

To find ∫ln(x)dx, we use integration by parts:

Let u = ln(x) and dv = dx

Then du = 1/x dx and v = x

∫ln(x)dx = uv - ∫v du

= x_ln(x) - ∫x * 1/x dx

= x_ln(x) - ∫1 dx

= x*ln(x) - x + C

The answer is xlnx – x + c.

Explanation

For matrices A × B ≠ B × A in general, so matrix multiplication is not commutative.

However, matrix addition is commutative and associative, and matrix multiplication is associative, but not commutative.

Explanation

The given partial differential equation:

∂²U/∂x² + ∂²U/∂y² = f(x,y)

is known as the Poisson equation.

If f(x,y) = 0, it would reduce to the Laplace equation.

Explanation

The Maclaurin series expansion of cos(x) is derived from its Taylor series expansion around x = 0:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

This series alternates signs and involves even powers of x, with factorials in the denominators.

The answer is cosx = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Explanation

Given y = x²/(1 + x²),

∫y dx = ∫[x²/(1 + x²)] dx = ∫[1 - 1/(1 + x²)] dx = x - arctan(x) + C.

Given ∫y dx = C + f, f = x - arctan(x).

x - 1 + 1/(x + 1)

Explanation

The derivative of 2^(-x) is -2^(-x) * ln(2), which can be written as -2⁻ˣ·log2 (considering log as ln or log base e implicitly in the derivative formula).

Explanation

The formula ∂f/∂x = lim(h → 0) [f(x + h, y) - f(x, y)]/h represents the partial derivative of f with respect to x, and it's evaluated at the point (x, y).

The answer is x at (x, y).

Explanation

Step 1: Evaluate the integral

∫(x² - x - 1) dx = (x³ / 3) - (x² / 2) - x

Step 2: Apply the limits of integration

[(1³ / 3) - (1² / 2) - 1] - [((-1)³ / 3) - ((-1)² / 2) - (-1)]

= [(1/3) - (1/2) - 1] - [(-1/3) - (1/2) + 1]

Step 3: Simplify the expression

= (1/3) - (1/2) - 1 + (1/3) + (1/2) - 1

= (2/3) - 2

= (2/3) - (6/3)

= -4/3