Things to know about or how to do on Exam 2:
Chapter 3
Section 3.1+3.2-Exponential fncts and logs
I)
The graphs of exponential and logarithmic functions
II)
How to shift/stretch their graphs (less important, but can be useful)
III)
Limits involving the exponential and logarithmic
IV)
Their asymptotes
V)
Solving eqns involving exponentials and logarithms using:
A. Laws of exponents
B. Expansion and contraction
C. Change of base formula
D. Check solutions in original eqn (logarithm equations)
E. Quadratic formula (should be memorized by now)
Section 3.2-Inverse functions
I)
Inverse function definition: (f-1(x) = y  f(y) = x)
II)
Use above defn to find inverse function
A. Ptwise
B. Functionwise (i.e. find an algebraic expression for the inverse)
III)
Domains and Ranges (know they switch)
IV)
Identify one to one functions
A. Table
B. Horizontal line test
C. Using basic definition algebraically (i.e. prove that whenever x1 ≠ x2
then f(x1) ≠ f(x2))
V)
Use reflection about y = x to find graph of f-1(x) from graph of f(x)
VI)
The inverse function derivative formula: d/dx(f-1(x))|x=b =
1/(d/dx(f(x))|x=f-1(b)) (using the formula involves 1) finding f-1(b), 2)
finding d/dx(f(x)), and 3) plugging those two into the formula)
Section 3.3-Exponential and logarithmic derivatives
I)
Derivatives for logs (d/dx(loga(x)) = 1/ln(a) 1/x)
II)
Logarithmic differentiation
III)
Derivative for exps d/dx(ax) = (ln(a))ax
IV)
Be able to use all these rules in conjunction with the rest of the
differentiation rules (power rule, product rule, chain rule, etc.)
Section 3.4-Exponential growth and decay
I)
Exponential growth and decay has
A. Constant relative growth rate
B. Solution y(t) = C0ekt (C0 is initial amount, k is relative rate)
C. d/dt(y(t)) = k y(t)
II)
Exponential growth and decay shifted (Newton’s law of cooling)
III)
Understanding them both and using them to solve word probs
IV)
Compound interest
Section 3.5-Inverse trig functions and derivatives
I)
Finding values of inverse trig fncts
A. E.g. sin-1(π/3)
B. E.g. sin-1(tan(π/3))
C. E.g. sin(x) if cos(x) = 4/5
II)
Domain restriction (know domain/range of sin-1, cos-1, tan-1)
III)
Inverse trig and trig properties
A. Cancellation
B. sin2(x)+cos2(x) = 1
IV)
Derivatives:
A. Use inverse definition with implicit differentiation to find derivatives
of inverse trig functions (sin-1(x) = y sin(y) = x…then implicitly
differentiate)
B. Memorize derivatives for sin-1, cos-1, tan-1
C. Use those memorized derivatives with other differentiation rules to
find more general derivative (e.g. d/dx(xcos-1(3-2x)+5x)
D. Find domains (less important but can be useful)
Section 3.6-Hyperbolic trig
I)
See list in email
II)
Know definitions of
A. Sinh and cosh in terms of exponentials
B. Tanh, csch, sinh, and coth in terms of sinh and cosh
III)
Evaluate hyperbolic fncts at given points
IV)
Evaluate their inverses
A. sinh-1(2)
B. sinh-1(tanh(4))
C. sinh(x) if cosh(x) = 5
V)
Be able to find their derivatives
A. Find d/dx(hyperbolic) using exponential defns
B. Find d/dx(hyperbolic) knowing d/dx(sinh(x)) = cosh(x) and
d/dx(cosh(x)) = sinh(x) (memorize these last two facts)
C. Find d/dx(inverse hyperbolic) using log expressions for the inverse
hyperbolics
D. Find d/dx(inverse hyperbolic) using implicit differentiation
VI)
Find log expression for inverse hyperbolic function
VII) Properties
A. cosh2(x)-sinh2(x) = 1
B. Use exponential definitions to prove simple properties
Section 3.7-L’hôpital’s rule
I)
Know L’hôpital’s rule:
lim
lim f ( x)
f ( x)

x  a g ( x) x  a g ( x)
lim
lim
if
f ( x)  0and
g ( x)  0
xa
xa
lim
lim
or
f ( x)  and
g ( x)  
xa
xa
II)
Know how to us e it, once, twice, maybe more
III)
Indeterminate forms:
A. Know the types
B. How to convert between them and eventually use L’hôpital’s
C. Identify even ifmixed in with other “normal” limits
IV)
Use “limit substitution”
Section 4.1-Max and Min values…be able to find:
I)
Local maxima and minima
A. On a graph
B. Using critical points
II)
Abs max and minima
A. On a graph
B. Using critical points and end points (when on a closed interval)
III)
Know when they don’t exist
IV)
Critical pts/numbers
Section 4.2-Mean value theorem
I)
Know mean value theorem (f’(c) = (f(b)-f(a))/(b-a)) with f continuous on
[a,b] and differentiable on (a,b).
II)
Know intermediate value theorem:
III)
Applications
A. Show equation has one and only one root (uses IVT and MVT)
B. Show average rate of change = instantaneous rate of change
somewhere
C. Find bounds on size of f at one end of an interval
D. Bound the size of a function
IV)
Know when MVT can be “violated”
Section 4.3-Derivatives and graphs
I)
f’
A. Increasing and decreasing
B. First derivative test to identify local max and min (f must be
continuous at that point)
C. Use tables to identify both A and B
II)
f’’
A. Concave up and down
B. Second derivative test for local max and min
C. Use table to identify
Section 4.4-Curve sketching
I)
Be able to find any and all of the following info about a fnct (when asked)
A. Domain
B. Intercepts
C. Symmetry
D. Asymptotes
E. Intervals of increase and decrease (f’)
F. Local max and min (f’)
G. Concave up and down (f’’)
H. Inflection points (f’’)
II)
Use this info to draw a graph that gets all this info correct
III)
(Be able to extend periodic functions as appropriate)
Section 4.5-Optimization
I)
Process
A. Read
B. Diagram
C. Introduce relevant variables
D. Find formula for main quantity
E. Use other relationships to express main quantity as a fnct of one
variable
F. Find abs max or min as requested including justifying
II)
Justification
A. Closed interval theorem
B. F’’>0 or <0 on entire interval
C. F’>0 to left of c and f’<0 to right of c for abs max (opposite for abs
min)
Section 4.7-Antiderivatives
I)
Given a function, find the antiderivative
A. Graphically
B. Algebraically
II)
Particle motion and acceleration
If in doubt, favor studying more of the recent material over the later material as the recent
material usually requires understanding of both.