Calculus I. When the following condition holds: if and only if

Transcription

1 Calculus I I. Limits i) Notation: The limit of f of x, as x approaches a, is equal to L. ii) Formal Definition: Suppose f is defined on some open interval, which includes the number a. Then When the following condition holds: if for every number > 0 there is a number > 0 such that if then Interpretation: This statement translates that if for every interval surrounding a value of x there exists a corresponding y interval of which the value of the f(x) remains in the interval, then the Limit is in the interval. Specifically the limit of f(x)=l. if and only if and Interpretation: This means that the limit must be the same from both sides of the function for the limit to be that value (and vice versa). Limit Laws: For c, constant, and the following limits exist: 1) 2) 3) 4) 5) 6) 7) 8)

2 9) 10) (for Real numbers) Direct Substitution: then If f is a polynomial or a rational function and a is in the domain of f, Continuity: i) ii) iii) Intermediate Value Theorem: If f is continuous on [a,b], then for all N in between f(a) and f(b) there exists a number c in (a,b) such that f(c)=n. L Hospital s Rule *Must have indeterminate form* II) Differentiation Derivative Differentiation Rules:

3 (Constant Multiple Rule) (Sum Rule) (Difference Rule) (Power Rule) (Constant Rule) (Product Rule)* (Quotient Rule)* (Chain Rule)* (Trig Rule)* (Trig Rule)*

4 (Trig Rule)* (Trig Rule)* *Special Techniques for Memorization* Product Rule: First times the derivative of the second plus second times the derivative of the first. Quotient Rule: Consider f(x) to be high and g(x) to be low. Now consider the D term to represent the derivative. We say: Low D-High minus High D-Low all over Low-squared Chain Rule: Derivative of the outer times the derivative of the inner **Trigonometry Rules** (aka: SST Rule ): First imagine whispering to somebody. We say psst. Now take off the p. We should have sst S S T Now draw in the appropriate functions: Underneath this line write the corresponding Co-Functions: Csc Csc Cot Now the last step we just remember the negative on lower middle. Thus the key to this technique is deriving the following: Csc -Csc Cot

5 Now to take the derivative we cross out the one we want to take and look right or left. Ex: Csc -Csc Cot Example 2: Csc -Csc Cot Example 3: Csc -Csc Cot Example 4: Csc -Csc Cot FOR MORE DERIVATIVES SEE LIST OF DERIVATIVES

6 Extreme Value Theorem: If f is continuous on a closed interval [a,b] then f has an absolute maximum and an absolute minimum somewhere in the interval. Fermat s Theorem: If f has local maximum or local minimum at a point x=d, then f (d)=0 provided f exists at that point. (In other words, to find a local maximum or minimum, the derivative must be zero at that point). Definition: A Critical Number of a function f is a number d in the domain of f such that either f (d)=0 or f (d) does not exists (DNE). (When we have a fraction for a derivative we set both the numerator and denominator equal to zero to find critical points). Local Max/Min: *If f has a local maximum or minimum at c, then c is a critical number of f.* Absolute Max/min: The Closed Interval Method: To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1) Find the values of f at the critical numbers of f in (a, b). 2) Find the values of f at the endpoints of the interval. 3) The largest of the values is the absolute maximum and the smallest is the absolute minimum. Rolle s Theorem: If f is a function and 1) F is continuous on the closed interval [a,b] 2) F is differentiable on the open interval (a,b) 3) F(a)=F(b) Then, there exists a number c in (a,b) such that f (c) =0.

7 Mean Value Theorem: If f is a function and 1) F is continuous on the closed interval [a,b] 2) F is differentiable on the open interval (a,b) Then there exists a number c in (a,b) such that (Where slope of the secant line is equal to the slope of the tangent line) Increasing/Decreasing Test: (a) If f (x)>0 on an interval, then f is increasing on that interval. (b) If f (x)<0 on an interval, then f is decreasing on that interval. The First Derivative Test: if f has a critical number c and is continuous: (a) If f changes from positive to negative at c, then f has a local maximum at c. (b) If f changes from negative to positive at c, then f has a local minimum at c. (c) If f does neither, then f has neither a local max nor local min. The Second Derivative Test: if f is continuous near c: (a) If f (c)=0 and f (c)>0, then f has a local minimum at c. (b) If f (c)=0 and f (c)<0, then f has a local maximum at c. Concavity Test: (a) If f (x)>0 for all x in some interval, then f is concave up on that interval (b) If f (x)<0 for all x in some interval, then f is concave down on that interval Inflection Points: A point P is an inflection point of f where the concavity changes; that is, where f goes from concave up to concave down, or from concave down to concave up. Newton s Method: Approximates the zeros of a function provided the derivative at the previous point is not zero. Converges with. *The initial point must be chosen near the actual zero. In other words, guess well. *

8