Flash Cards: Limits and Derivatives

A limit is the value that a function approaches as the input approaches a certain point. It is denoted as lim (x→a) f(x) = l, meaning f(x) approaches l as x approaches a.

Average velocity between t₁ and t₂ is given by: v_avg = (s(t₂) - s(t₁)) / (t₂ - t₁), where s(t) is the distance function.

The derivative of a function at a point measures how the function's output changes as the input changes. It is the slope of the tangent line at that point.

The derivative of s = 4.9t² with respect to t is ds/dt = 9.8t, which represents the instantaneous velocity at time t.

To compute limits, substitute values of x approaching a in f(x) from both sides. Check if the left-hand limit and right-hand limit are equal to establish lim (x→a).

The right-hand limit, written as lim (x→a⁺) f(x), is the value that f(x) approaches as x approaches a from the right.

The left-hand limit, noted as lim (x→a⁻) f(x), is the value that f(x) approaches as x approaches a from the left.

If the left-hand limit and right-hand limit at a point are not equal, the overall limit does not exist at that point.

The limit of a constant function f(x) = c as x approaches a is simply c, i.e., lim (x→a) c = c.

lim (x→2) f(x) = 3*2 = 6.

As x approaches a point, if the function's values get closer to a certain number, that number is the limit at that point.

The derivative gives us the rate of change of a function, indicating how steeply it rises or falls at a specific point.

lim (x→0) f(x) = 0, since f(x) approaches 0 as x gets closer to 0.

A function limit defines the behavior of the function near a point rather than at that point itself.

Yes, a function can have a value at a point where the limit does not exist, such as jump discontinuities.

lim (x→0) |x| = 0, as values from both sides approach 0.

Derivatives are used to find the instantaneous rate of change, such as velocity in physics when studying motion.

The slope of a tangent line to the graph of a function at a point represents the derivative of the function at that point.

Limits can be interpreted visually as the value the graph approaches as x gets very close to a specific point.