f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.

True, as the limit of f'(x) as x approaches 0 exists and equals 0.

f(x) is a polynomial function, hence it is differentiable everywhere including at x = 1.

f'(x) = 3x^2 - 3 = 0 gives x = ±1, thus critical points are x = -1 and x = 1.

f'(x) = 3x^2 - 3, thus f'(1) = 0.

Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.

The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.

The function is not continuous at x = 0 since the limit does not equal f(0).

To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.

For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.

For continuity at x = 1, both pieces must equal 4, hence the function is continuous.

To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.

Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.

To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.

The function is not continuous at x = 1 because the left-hand limit does not equal the function value.

To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.

To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.

For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.

Both conditions must hold true for continuity at x = 2.

To be continuous at x = 2, k must equal f(2) = 2^2 = 4.

The function |x - 3| is continuous everywhere, including at x = 3.

f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.

The equation x^2 + y^2 + Dx + Ey + F = 0 represents a family of circles.

The equation y^2 = 4ax represents a parabola.

The equation y^2 = 4ax represents a parabola that opens to the right.

The equation y = a^x represents an exponential function where a is a constant.

The gravitational field inside a uniform spherical shell is zero.

V = -g * r = -9.8 N/kg * 6.4 x 10^6 m = -62.72 x 10^6 J/kg.

The gravitational potential at the surface of the Earth is negative, approximately -9.8 J/kg.

The gravitational force acting on a satellite depends on the mass of the satellite, the mass of the Earth, and the distance from the Earth according to Newton's law of gravitation.