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