Calculus Lecture Notes

This set of flashcards covers key concepts and problems related to limits, continuity, derivatives, and the properties of functions highlighted in the lecture notes.

What is the limit of the function as x approaches 0 for the function √(x+1) - 1 when evaluated using the values in the provided table?

The limit is approximately 0.5000.

What can be concluded about the limit of f(x) as x approaches -2 from both sides?

The left-hand limit approaches negative infinity and the right-hand limit approaches positive infinity, indicating a vertical asymptote.

What conditions must be met for a function f(x) to be continuous at a point c?

1. f(c) is defined. 2. The limit as x approaches c exists. 3. The limit as x approaches c equals f(c).

Identify the type of discontinuity at x = -3 for the function f(x) = (x + 3) / (x² - 2x - 15).

The discontinuity at x = -3 is removable.

What is the derivative of the function f(x) = x² + x - 9 using the limit process?

f'(x) = 2x + 1.

Find the derivative of the function y = cos(8x).

y' = -8sin(8x).

What is the difference between a relative maximum and an absolute maximum?

A relative maximum is the highest point in a local vicinity, while an absolute maximum is the highest point over the entire domain.

For the function f(x) = -5x³ + 15x, what are the critical numbers?

The critical numbers are x = -1 and x = 1.

Determine the open intervals on which f(x) = x² - 5x³ is concave upward.

Concave upward on the intervals where the second derivative is positive.

What must be true for limits to exist and be equal at a point?

Both the left-hand limit and the right-hand limit must be equal and finite.