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# Criswell Calculus

### AP Calculus Theorems and Concepts "Limits" (Chapter 1)

TermDefinition
Definition: The Intermediate Value Theorem (IVT) If "f" is a continuous function on the closed interval [a,b] and K falls between f(a) and f(b), then there must exist at least one c on the open interval (a,b) where f(c) =k
Concept: Continuity of a function When considering a specified domain for the function. If no holes, gaps, or asymptotes occur the function is said to be this on the interval. The idea is that with the set boundaries the function would "hold water".
Concept: Fundamental Theorem of Algebra The intuitive understanding of how the degree of polynomials influence both the graph and the number of solutions for a given function. An example of the might be y = x^3 –7x^2+9x–2, where there could be up to 3 roots and at most 2 "bends" in the curve.
Definition: Removable Discontinuity When a function "f" is defined at every point on an interval except for one value, say "c". At this location the Lim as x approaches from both the right and left equals some specific value such as y=12 but f(c) ≠ 12. "Concept of hole in graph"
Definition: Infinite Discontinuity When a restricted value for a function creates asymptotic behavior diverging to positive infinity on one side and negative infinity on the other. This is said to occur.
Definition: Jump Discontinuity When the limit as x approaches a certain value, say "a", is different but specified such as y=3 from the right and y = –7 from the left, this is said to be the result.
Definition: Average Rate of Change As a formula: {F(b) –F(a)} / (b –a), which would be the same as the definition of slope m = (y2 –y1) / (x2 – x1).
Conceptual Definition: Limit Let "f" be a function defined at every point "c" on the open interval (a,b) except possible at "c" itself, If there exists a real number L such that the limit as x approaches c from both the right and left, then we say the limit exists.
Three ways in which limits do not exist... "DNE" 1) Jump, Step, or Gaps 2) Oscillating Functions 3) Unbounded Behavior Be able to supply examples of each with appropriate limits when asked.
What are the techniques for evaluating Limits? 1) Crude Method: Table of values or Graph 2) Direct Substitution 3) Factoring Expression 4) Using common denominators 5) Conjugates 6) Trig Theorems
Formal Definition of Continuous: A function "f"is said to be this if for every c that is an element of ( a , b ) there exists a limit as x approaches c from both the right and left to which equals f (c).
Factor Difference of Cubes: A^3 – B^3 (A–B)(A^2+AB+B^2)
Factor Sum of Cubes: A^3 + B^3 (A+B)(A^2–AB+B^2)
Sum formula for sine sin (x + y) = sin(x)cos(y) + sin(y)cos(x)
Difference formula for sine sin (x – y) = sin(x)cos(y) – sin(y)cos(x)
Sum formula for cosine cos (h + k) = cos(h)cos(k) – sin(h)sin(k)
Difference formula for cosine cos (h – k) = cos(h)cos(k) + sin(h)sin(k)
Double angle formula for sine sin (2u) = 2 sin(u)cos(u)
Double angle formula for cosine cos(2w) = cos^2 (w) –sin^2 (w) cos(2w) = 2 cos^2 (w) – 1 cos(2w) = 1 – 2 sin^2 (w)
Pythagorean Identities for Trigonometry sin^2 (g) + cos^2 (g) = 1 1 + cot^2(j) = csc^2 (j) tan^2 (m) + 1 = sec^2 (m)
Trigonometry Limit Theorem for sine lim t–>0 sin (t) /t = 1
Trigonometry Limit Theorem for cosine lim r–>0 (1–cos (r) )/ r = 0
Created by: Troy.Criswell
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