calc final s1

What is limit notation used for in calculus?

To express what f(x) approaches as x approaches a value a, not necessarily the actual value of f(x).

What are the ways to find limits?

Plug in (direct substitution), Simplify (factor & cancel), Rationalize, Use a graph.

When does a limit exist?

A limit exists if the left-hand limit equals the right-hand limit.

What indicates a vertical asymptote?

A vertical asymptote occurs when \lim_{x \to a} f(x) = \pm\infty and the denominator equals zero without canceling.

What is the condition for a horizontal asymptote?

A horizontal asymptote occurs if \lim_{x \to \pm\infty} f(x) = L.

What defines continuity at x=a?

A function is continuous at x=a if f(a) exists, \lim_{x \to a} f(x) exists, and the limit equals the function value.

What are the indicators that a function is not continuous?

Holes, jumps, and asymptotes.

What is the Squeeze Theorem?

If a function is squeezed between two functions that have the same limit, then it has that limit as well.

What does differentiation measure?

The derivative measures the slope of the tangent line and the instantaneous rate of change.

What is the limit definition of the derivative?

f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}.

What does it mean for a function to be differentiable?

A function is differentiable at a point if it is continuous at that point; however, continuous functions may not be differentiable.

What are some cases where a function is not differentiable?

At corners, cusps, vertical tangents, or discontinuities.

What does the Power Rule state?

If f(x) = x^n, then f'(x) = n*x^{n-1}.

How do you find a tangent line to a function?

You need the slope (which is the derivative) and a specific point on the function.

What does the First Derivative Test involve?

Analyzing the sign of the first derivative to determine local maxima and minima.

What does the Second Derivative Test indicate?

If f'(a)=0 and f''(a)>0, then there's a local min; if f'(a)=0 and f''(a)<0, then there's a local max.

What are inflection points?

Points where the concavity of a function changes, occurring where f''=0 or undefined and the sign changes.