Explanation

The exponential function is its derivative and integral.

The integral of eˣ concerning x.

- The integral of eˣ is eˣ.  

- Don't forget the constant of integration C.

Apply the integration rule for eˣ.

Apply the integration rule:  

∫ eˣ dx = eˣ + C

The integral of eˣ is eˣ + C.

Explanation

The sound level β (in dB) is given by:

β = 10 log(I / I₀)

where I₀ is the threshold intensity (faintest audible sound). Given I = I₀:

β = 10 log(1) = 0 dB

Explanation

In the quadratic equation ax² + bx + c = 0, a cannot be zero.

If a = 0, the equation becomes linear (bx + c = 0), and it's no longer a quadratic equation.

Explanation

For matrices A and B, the inverse of the product AB is given by:

(AB)⁻¹ = B⁻¹A⁻¹

Explanation

The derivative of y = x² is dy/dx = 2x.

At x = 2, the slope is 2(2) = 4.

The tangent line equation is y - 4 = 4(x - 2), which simplifies to y = 4x - 4, or 4x - y = 4.

Explanation

To determine the type of conic section, rewrite the equation in standard form:

9x² + 4y² = 36

Divide by 36:

x²/4 + y²/9 = 1

Since 9 > 4, the major axis is vertical.

Explanation

A critical point occurs where the derivative of a function is zero or does not exist.

These points are important for finding local maxima, minima, or points of inflection.

Explanation

The ratio test and root test are inconclusive when the limit of the ratio (or root) of successive terms is exactly 1.

These tests fail to determine convergence or divergence in this case.

Explanation

A quadratic equation has the form ax² + bx + c = 0, and it typically has two roots. 

These roots can be real or complex, and they can be distinct or repeated (in the case of a double root).

Explanation

A matrix has an inverse only if its determinant is non-zero, meaning it is non-singular.

Singular matrices (with determinant = 0) do not have an inverse.