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AP Calculus AB Semester 1
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Gravity
Key Concepts:
Terms in this set (66)
What are the three steps to solving optimisation problems
1. Find the formula for the function you are trying to minimise/maximise
2. If necessary, eliminate a variable so there is only one (use info from question)
-Find the domain
3. Use calculus to find maximum or minimum and prove it
d/dx cscx
-cscxcotx
d/dx cosx
-sinx
d/dx tanx
sec²x
Relative/local max/min definition
Highest/lowest point if you look immediently to the left and right
A function is not differentiable if... (3)
1. You can't find a tangent line
2. If it's not continuous
3. If the slope is undefined
Continuity definition
An unbroken, consistent graph with no holes or jumos
n/0 is a...
Vertical asymptote
You can find the absolute extrema by...
Finding the highest and lowest point in a graph
d/dx secx
secxtanx
d/dx cotx
-csc²x
d/dn y=e²ⁿ
y'=2e²ⁿ
d/dn y=eⁿ
y'=eⁿ
How do you find v(t) and a(t) when given s(t)
v(t)=s'(t)
a(t)=s''(t)
d/dx lnx
1/x
If v(t) and a(t) have the same sign, they are...
Speeding up
If v(t) and a(t) have different signs they are...
Slowing down
What is the value when in lim (x→∞) the top exponent is greater than the bottom?
It DNE
Steps to solve related rates problems(3)
1. Write down what you know and draw a diagram
2. Write down what you want to know
3. Find a formula that relates the functions, differentiate, and solve
d/dn aⁿ
aⁿ ln(a)
d/dn log aⁿ
1/ln(a)
0/0 is a...
Hole
d/dx a (a is a constant)
0
3 steps to find a limit
1. Plug in the value
2. Simplify and try again
3. Plug in two numbers really close
d/dx xⁿ
nxⁿ⁻'
Absolute/global max/min definition
Highest/lowest overall value
How do you find relative min?
If f'(x) changes from -→+
How do you find relative max?
If f'(x) changes from +→-
f''(x) changes sign if...
f(x) changes concavity
d/dx cos⁻' x
- ____1___
(√1-x²)
What is the range of a function?
Valid inputs of the function on the y axis
d/dx sin⁻' x
____1___
(√1-x²)
f''(x) is 0 if...
f(x) is linear
f''(x) if negative if...
f(x) is concave down
f''(x) is positive if...
f(x) is concave up
How do you find average slope?
___f(b)-f(a)___
b-a
How to do implicit d/dx?
When y cannot be on one side, treat y as a function and solve for y'
When you are finding the derivative, you are finding the______ of f(x)
Slope
What is the chain rule?
f(x)=f(g(x))
f'(x)=f'(g(x))•g'(x)
d/dx Multiplication
Left d-right + right d-left
Quotient rule
___low d-high - high d-low___
(bottom)²
d/dx sinx
cosx
How do we know it's a critical/stationary point?
If f(x) equals 0 or DNE
What is the acceleration function?
Whether you are speeding up or slowing down
What is the velocity function?
+=moving right -=moving left
Rate of change of the position in a specific direction (m/s)
What is distance?
A total number of units one moved throughout their journey
What is displacement?
The distance from where you finished a journey from where you started
d/dx tan⁻' x
____1____
1+x²
What is the value when in lim(x→∞) is greater than the top?
It equals 0
How to find a derivative of a function of a certain x value on calculator?
Math→8
How to find x intercepts of a function on calculator?
Window(adjust to domain)→y=(type in equation)→graph→
2nd→trace→enter on left of x intercept→enter on right of x intercept→enter
How to find the points of intersection on the calculator?
Adjust window→graph→2nd→trace→5→enter→enter→ move it close to the intersection point and press enter
What shows us points of inflection?
f''(x)
What does f'(x) show us?
Rate of change.
f'(x) is above the x axis if...
The slope of f(x) is increasing
f'(x) is below the x axis if...
The slope of f(x) is decreasing
What is the equation of point slope form?
y-b=SLOPE(x-a)
How do you find the slope for a graph if you know the equation?
Plug in x to the derivative function and leave it as is
Mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)
f '(c) = [f(b) - f(a)]/(b - a)
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
d/dx arcsecx
1/ |x|√x^2-1
Definition of Derivative
lim f(x+h)-f(x)/h
h->0
Alternate definition of derivative
lim x->a f(x)-f(a)/x-a
d/dx [f(x) + g(x)]
f'(x) + g'(x)
lim x->0 sinx/x
=1
lim x->0 (1-cosx)/x
0
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