3.1-4.5 Test Flashcards
What is the quadratic formula?
x=(-b±√(b^2 - 4 ac)/(2a)
E=?
lim = (1+(1/n))^n
n->∞
What is a Factorial? What’s the Factorial of 4? (or 4!)
The product of all positive integers less than or equal to a given positive integer
4! = 1 * 2 * 3 * 4 = 24
(d/dx) Sin^-1x
1/
√(1-x^2)
(d/dx) Cos^-1x
-1/
√(1-x^2)
(d/dx) Tan^-1x
1/
(1+x^2)
(d/dx) Csc^-1x
-1/
x√(x^2 -1)
(d/dx) Sec^-1x
1/
x√(x^2 -1)
(d/dx) Cot^-1x
-1/
(1+x^2)
When Solving for Related Rates what 3 steps do you take?
- Identify what Rates, Lengths, Relations, and variables are given
- Find an equation that involves the unknown Rate(s) with the known Rate(s) and Given Variables
- Solve for the unknown Rate algebraically with respect to time (or d/dt)
What 3 equations do you use when given a starting and end Exponential Growth/Decay?
P(t) = P0e^(kt)
Pf = Final = P0e^Final*k
k = 1/t ln(Final/P0)
What is the PROPER Orientation of the Derivative Product rule?
f’(x) * g(x) + f(x) * g’(x)
What is the PROPER Orientation of the Quotient rule?
f’(x) * g(x) - f(x) * g’(x)
/g(x)^2
(d/dx) Sinhx
e^x - e^-x
/2
(d/dx) Coshx
e^x + e^-x
/2
(d/dx) Tanhx
Sinh
/Cosh
(d/dx) Cothx
Coshx
/Sinhx
(d/dx) Sechx
2
/Coshx
(d/dx) Cschx
2
/Sinhx
Intermediate Value Theorem
F(a) < K < F(b)
a < c < b
F(c) = K
Mean Value Theorem
If f is Cont. on [a,b]
& f is Diff. on (a,b)
Then there exists a c s.t.
f’(c) = f(b) - f(a)
/b-a
Cos(0, 2pi)
1
Cos(pi)
-1
Cos(pi/2)
0
Sin(0,pi)
0
Sin(pi/2)
1
Sin(3pi/2)
-1
A graph of Sinx starts at?
0 Because there’s no Zink
A graph of Cosx starts at?
1 Because co-friends are +1
What does the 1st Derivative Test tell us?
If f’ changes from + to - at c, then f has a local maximum at c.
If f’ changes from - to + at c, then f has a local minimum at c.
If f’ is + to the left and right of c, or - to the left and right of c,
then f has no local maximum or minimum at c.
What does the 2nd Derivative Test tell us?
Suppose f’’ is continuous near c.
If f’(c) = 0 and f’‘(c) > 0, then f has a local minimum at c.
If f’(c) = 0 and f’‘(c) < 0 then f has a local maximum at c.
How do you prove that there are only so many solutions to an equasion?
You prove by contradiction, via IVT or MVT to disprove that there could be more solutions
L’Hopital’s Rule?
If a limit function is ∞/∞ or 0/0 you may take the derivative of the top and bottom in order to rewrite the equation so it’s more solvable
Graphing a Curve: Step 1
Finding the Domain
Where x is defined
Graphing a Curve: Step 2
Intercepts
x & y
Graphing a Curve: Step 3
Symetry
Graphing a Curve: Step 4
Asymptotes
Graphing a Curve: Step 5
Use the numero line test to see if intervals are increasing or decreasing
Graphing a Curve: Step 6
Derive and find Critical Numbers (#’s or c where f(c) = 0 or is undefined)
Graphing a Curve: Step 7
Use the 1st & 2nd Derivative test and critical numbers to identify Local/Global Minimums/Maximums
Graphing a Curve: Step 8
Identify Concavity/Inflection Points
What’s the base 3 Indeterminate forms?
∞/∞, 0/0, ∞/0
Assuming that the Lim x-> 0+ , What would the result of the Indeterminate Product x^2 * 1/x?
Lim x-> 0+ x = 0
Assuming that the Lim x-> 0+ , What would the result of the Indeterminate Product x * 1/x^2?
Lim x-> 0+ x =∞
Assuming that the Lim x-> 0+ , What would the result of the Indeterminate Product x * 1/x?
Lim x-> 0+ x =1
How would you turn an Intermediate Product into a Base Intermediate Form?
f(x) * g(x) = f(x)/[1/g(x)]
What 3 steps should you take to turn an Intermediate Difference into a Base Intermediate Form?
- Try to find a common denominator or common factor that can be factored
- Rearrange algebraically so L’H’s Rule can be used
- Use L’H’s Rule
Tan(pi,0)
0
Tan(pi/2)
Undetermined/V.Asym
ln(1) = ?
0
ln(0) = ?
Indeterminate
ln(e) = ?
1
ln(e^x) = ?
x
What do you set to what to find a Y-Intercept?
You set f(x) to 0;
f(0)
What do you set to what find a X-Intercept?
You set x to 0
x = 0
In what ways could you identify if there could be a Horizontal Asymptote?
- If Lim(x->±∞) = L then y = L is a Horizontal Asymptote
- Divide by the highest power of x
How do you identify if there could be a Vertical Asymptote?
If Lim(x->a±) = ±∞ then x = a is a Vertical Asymptote
What functions can you use to check if a graph is symetrical?
f(-x) = f(x)
f(-x) = -f(x)
f(x+p) = f(x) (periodic symmetry)
