Semester Review

1
Semester Review

• Ch 1-3

2
Limits graphically

• Near not at

3
Limits algebraically

• Simplify then try again

4
Limits LHSRHS

• Lhs vs Rhs notation and lim existence

5
Continuity limf(a)

• 3 conditions
• No breaks/holes

6
HA and VA

• HA lim x-gt inf
• Divide by highest power
• VA denom 0 after cancel
• Hole if it canceled

7
lim defn of derivative
8
Equation of tan line

• y-f(x0) f '(x0)(x-x0)

9
f, f graphs

• f tells the slope of f at any x
• Look for horizontal tangent lines

10
Dx of power and trig

• Multiply by power
• Decrease the power by one
• Memorize trig derivatives

11
Dx quotient, product

• Recognize WHEN they are needed
• AB AB

12
Dx chain

• Der outside, keep inside der inside
• f (g(x))g(x)

13
Implicit

• Tack on dy/dx when diff y
• Diff both sides
• Collect dy/dx terms, factor, and solve

14
Related Rates

• Equation
• Amount
• Rates
• Differentiate and tack on rates, e.g. da/dt

15
MVT

• Avg ROC Inst ROC

16
Rel max/min

• f (c)0
• 1st der test Test left and right of c into f
• Inc then dec
• 2nd der test Test cp into f
• Concave up

17
Inflection points

• f (d) 0
• Plug in left and right of d into f to
  determine concavity
• Inflection points are where f changes concavity

18
Sketching graphs

• Function values
• Horizontal tangent lines
• Regions of Increasing/Decreasing
• Regions of Concavity

19
Differentials

• dy f (x0)dx
• Shows approximate change (dy) in y value ?y by
  riding the tangent line (f (x0)) a little bit
  away (dx) from the point of tangency (x0,f(x0))

20
Optimization

• Generally two equations
• Constraining equation
• Equation to optimize
• Plug constrained eqn into optimize eqn
• Take derivative 0
• Test if max/min
• Allow possibility of endpoints

21
Reading f ' graphs

• f
• Inc
• Dec
• C up
• C dn
• Cp
• Loc max

• f '
• Pos -Above
• Neg -Below
• Inc
• Dec
• 0 on x axis
• gt -

• f ''
• Pos
• Neg
• Neg