Calculus BC

• Definition of limit: The limit as x approaches c of f of x exists if each of the one-sided limits respectively exist and are the same
• Reasons for limit non-existence:

  • One-sided limits disagree, leading to a jump discontinuity
  • Unbounded behavior, such as a vertical asymptote
  • Oscillation, where the function wiggles back and forth and does not approach a singular value
• Linearity of limits:

  • The limit of the sum is the sum of the limits
  • Multiplying by a constant works as expected
• To simplify rational expressions to lowest terms, factor the numerator and denominator, then cancel common factors
• When multiplying fractions with just numbers, multiply the numerators and denominators together
• When dividing fractions, multiply by the reciprocal of the fraction in the denominator
• To add or subtract rational expressions, find a common denominator by factoring the denominators and then rewrite each fraction in terms of the least common denominator
• The average rate of change for a function on the interval from a to b is the slope of the secant line between the two points A, F of A and B, F of B, given by the formula: (f(b) - f(a)) / (b - a)
• The difference quotient represents the average rate of change of a function on the interval from x to x + h, and is calculated as (f(x + h) - f(x)) / h
• To calculate the difference quotient, substitute x + h into the function to find f(x + h), subtract f(x), simplify the numerator, and then divide by h
• In the context of limits, as x approaches a certain value, f(x) approaches a specific value
• For piecewise defined functions, consider different cases for the function's behavior based on the input values
• When x is near a specific value but not equal to it, the limit of f(x) as x approaches that value can be determined
• In the language of limits, the limit as x approaches one of F of x is equal to L
• The value of F at one is actually equal to zero, not L
• The limit as x goes to a of f of x equals L means that f of x gets arbitrarily close to L as x gets arbitrarily close to a
• For any function f of x and for real numbers a and L, the limit as x goes to a of f of x equals L means that f of x gets arbitrarily close to L as x gets arbitrarily close to a
• The limit of f of x as x goes to one is two, even though f of one itself does not exist
• The limit as x goes to two of g of x does not exist
• The limit as x goes to negative two from the right of h of x is infinity
• The limit as x goes to negative two from the left of h of x is negative infinity
• The limit as x goes to negative two of h of x does not exist
• The limit as x goes to zero of sine pi over x does not exist
• The limit as x goes to zero of x squared sine one over x does not exist
• The limit laws apply if the limits of the component functions exist as finite numbers
• The limit of the sum f of x plus g of x is equal to the limit of f of x plus the limit of g of x
• The limit of the difference f of x minus g of x is equal to the limit of f of x minus the limit of g of x
• The limit of C times f of x is C times the limit of f of x
• The limit of the product f of x times g of x is the product of the limits
• The limit of the quotient f of x divided by g of x is the quotient of the limits, provided that the limit of g of x is not equal to zero
• The limit as x goes to two of x squared plus 3x plus six divided by x plus nine is 16/11
• The squeeze theorem states that if f of x is less than or equal to g of x, which is less than or equal to h of x, and the limits of f of x and h of x as x approaches a are equal, then the limit of g of x as x approaches a is also equal to that limit
• Product rule only applies when the component limits both exist
• Product rule does not provide information if the second limit doesn't exist
• Squeeze theorem can be used when a function is trapped between two other functions with the same limit
• Example of using algebra to find bounding functions for a function trapped between x squared and -x squared
• Glacial erosion is the breaking down and removal of rocks and sediment by natural forces
• In glacial environments, the 2 main forms of erosion are abrasion and plucking
• Abrasion:

  • A sandpapering effect caused by small rocks embedded within the glacier rubbing on bedrock
  • Usually leaves a smooth surface with scratches called striations
• Plucking:

  • Meltwater from glaciers freeze around broken or cracked parts of rock, breaking it off from the bedrock or sides as the ice moves down the slope
  • Most prominent when there are many joints in the rock, as water can penetrate the rock and freeze in the cracks