Exam 3 Flashcards
Linear Approximation Formula
L(x)=f’(a)(x-a)+f(a)
Differential f(x+∆x)
~f(x)+f’(x)∆x
Demand (price)
P(x)
Revenue R(x)
=xp(x)
Cost
c(x)
Profit
P(x)=R(x)-C(x)
or xp(x)-C(x)
Critical Numbers
In domain, when f’(x)=**0 **(Horizontal Tangent Lines)
When f’(x)=undefined (Vertical Tangent Lines or cusps)
Increasing/Decreasing Function
f’(x)>0 increasing
f’(x)<0 decreasing
Local Max
increases then decreases
Local Min
decreases then increases
First Derivative Test
Set top and bottom of derivative=0. Find critical numbers. Make number line. Find max and min based on changes in sign across interval. Remember domain!
Plug back in to find actual min/max value of function.
Absolute Extrema
Find the highest (max) and lowest(min) with first derivative test critical points and using the endpoints of the interval. Plug all these x values into f(x).
Mean Value Theorem
if a function is continuous on [a,b] and differentiable on (a,b), then there exists some value x=c in (a,b) such that f(b)-f(a)/b-a = f’(c).
Take f(b)-f(a)/b-a and set equal to the derivative (in derivaitve use x or c). Solve for that variable (x or c)
Rolle’s Theorem
If a function is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then there exists some value x=c in (a,b) such that f’(c)=0
Take the derivative and set it equal to 0. Solve for x, that is your c value.
Second derivative and concavity
Convae up is when f’‘(x)>0
concave down is when f’‘(x)<0
Inflection point
Where concavity changes
Set f’‘(x)=0 and test points on the number line.
Second derivative test
Take first derivative. Set f’(x)=0 for critical numbers. Plug these critical numbers into f’‘(x)
f’‘(x)>0 you have a minimum.
f’‘(x)<0 you have a maximum.
f’‘(x)=0 then can’t be determined by second derivative test.
Domain of an even root
inside>or equal to 0
Domain with a denominator
can’t equal zero
logs/ln domain
inside> 0
X intercept
set y =0
y intercept
set x=0
Vertical asymptotes
set denominator=0 and solve after you simplify.
x=a is the form.
Horizontal asymptotes
look at the powers
powers are the same, HA is the ratio of coefficients
Bottom is bigger HA y=0
Top is bigger, no HA
Cusp Vs. VTL
cusp is if the function changes signs across the first derivative at that point. VTL and not a cusp if it does not change signs across the first derivative at that point.
multiplicity
even powers- same sign; odd powers change sign. If raised to a fraction, use the upper number. E.g. 4/3 is even.
LHopital’s Rule indeterminate forms
0/0 or infinity/infinity
If it’s a different indeterminate form for lhopital
rewrite the function.
lhopitals with a function raised to a function
raise it to e^ln(function).
move e to the outside so the whole limit is raised to e. Solve the ln limit, that answer is the power on the e. Simplify.
Optimization
identify the primary equation. Identify the constraint. isolate a variable in the constraint and plug it in to the primary equation. Take derivative of primary and set it equal to 0 to find the value’s max or min point. Find the other variable. Plug both back in for the acutal max or min.
