Calculus 1.1: Review of Trig/Precal A. Lines 1. Slope: 2. Parallel lines—Same slope Perpendicular lines—Slopes are opposite reciprocals 3. Equations of lines: point-slope form:y – y 1 = m(x – x 1 ) slope-intercept form:y = mx + b standard form:Ax + By = C

B. Functions 1. Function (from set D to set R)—a rule that assigns a unique element in R to each element in D 2. Domain & Range intervals 3. Symmetry: even function if f(-x) = f(x) odd function if f(-x) = -f(x) 4. Piece-wise functions 5. Composite functions:

C. Inverse functions: 1. f is one-to-one if Graphs of inverse functions are reflections across the line y = x 4. To find an inverse function, solve the equation y = f(x) for x in terms of y, then interchange x and y to write y = f -1 (x)

D. Exponential & Logarithmic Functions 1. Exponential function: 2. Logarithmic function: E. Properties of Logarithms:

F. Trigonometry Review 1. Trig Functions: 2. Remember: Special Right Triangles!!

3. Trig Graphs: a. Periodicity: b. Even/Odd: c. Variations: y = a sin (bx – c) + d

4. Inverse Trig Functions: Remember: Keep Calculators in Radian Mode!!

Calculus 1.2: Limit of a Function A. Definition: Limit: “The limit of f(x), as x approaches a, equals L”—if we can make f(x) arbitrarily close to L by taking x sufficiently close to a (on either side of a) but not equal to a. Ex 1: see fig 2 p.71 (Stewart)

B. One-sided limits: (from the left) iff C. Estimating Limits using (from the right) Note:

D. Limit Laws: (if c is a constant and and exist) 1. Sum Rule: 2. Difference Rule: 3. Constant Multiple Rule:

4. Product Rule: 5. Quotient Rule: 6. Power Rule: ( n is a positive integer ) 7. Root Rule: (n is a positive integer)

E. Direct Substitution Property: If f is a polynomial or a rational function and a is in the domain of f, then:

Calculus 1.3: Limits Involving Infinity A. Definition: (Let f be a function defined on both sides of a) means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a (but not equal to a) Note: arb. large negative

B. Definition: The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true:

C. Definition: Let f be a function defined on the interval Then means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Note: taking x large neg.

D. The line y = L is called the horizontal asymptote of y = f(x) if either: or E. Theorem: if r > 0 is rational, then and

F. Method for finding limits at infinity: 1. Divide top and bottom of rational function by the largest power of x in the denominator 2. Simplify using theorem in E above

Calculus 1.4: Continuity A. DEF: A function f is continuous at a number a if (assuming f(a) is defined and **Remember, this means:

B. Types of Discontinuity 1. Removable 2. Infinite 3. Jump 4. Oscillating see fig 2.21 p. 80

C. Continuous Functions 1. A function is continuous from the right at a if 2. A function is continuous on an interval [a,b] if it is continuous at every number on the interval 3. The following are continuous at every number in their domains: polynomials, rational functions root functions, trig functions

D. Intermediate Value Theorem: Suppose f is continuous on [a,b] and f(a) < N < f(b) Then there exists a number c in (a,b) such that f(c) = N

Calculus 1.5: Rates of Change A. Average Rates of Change 1. Average Rate of Change of a function over an interval – the amount of change divided by the length of the interval 2. Secant Line – a line through 2 points on a curve

B. Instantaneous Rates of Change 1. Tangent Lines The tangent line to y = f(x) at the point P(a,f(a)) is the line through P with slope: or (if the limit exists)

2. Velocities Instantaneous velocity v(a) at time t=a:

Calculus 1.6: Derivatives A. Definitions: 1. Differential Calculus—the study of how one quantity changes in relation to another quantity. 2. The derivative of a function f at a number a: (if the limit exists), or

B. Interpretation of derivatives 1. The tangent line to y = f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is f ’(a) 2. The derivative f ’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a

C. The Derivative as a Function 1. Definition of the derivative of f(x) as a function: Ex: Find f ‘ (x):

Calculus 1.7: Differentiability A. Other Notation for Derivatives: B. DEF: A function f is differentiable at a if f ‘(a) exists. It is differentiable on (a,b) if it is differentiable at every number in (a,b). Ex 1:

C. Cases for f NOT to be differentiable at a: 1. Corner – one-sided derivatives differ 2. Cusp – derivatives approach from one side and from the other 3. Vertical Tangent – derivatives approach either or from both sides 4. Discontinuity – removable, infinite, jump or oscillating Ex 2: Find all points in the domain where f is not differentiable. State which case each is:

D. Graphs of f ’ 1. Sketching f ’ when given the graph of f see Stewart p.135 fig 2 a. p. 106 #22 b. p. 105 #13-16 c. p # Sketching f when given the graph of f ’ a. p. 107 #27,28

Calculus: Unit 1 Test Grademaster #1-40 (Name, Date, Subject, Period, Test Copy #) Do Not Write on Test! Show All Work on Scratch Paper! Label BONUS QUESTIONS Clearly on Notebook Paper. (If you have time) Find Something QUIET To Do When Finished!