Studied by 3 people
Unit 1: Limits and Continuity (1-36) Unit 2: Differentiation: Definition and Fundamental Properties (37-68) Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (69-80) Unit 4: Contextual Applications of Differentiation (81-101) Unit 5: Analytical Applications of Differentiation (102-126) Unit 6: Integration and Accumulation of Change (127-148) Unit 7: Differential Equations (149-154) Unit 8: Applications of Integration (155-164)
What is a limit?
the value that a function approaches as the variable within the function gets nearer to a particular value
What do we focus on when it comes to limits?
what’s happening around the point, not the point itself
How do you find the limit of a simple polynomial?
plug in the number that the variable is approaching
When does a limit not exist?
if the graph approaches two different values for the same number
What are ways to find limits?
look on the graph to see what value it approaches
estimate from a table
algebraic properties
algebraic properties
What are algebraic properties for finding limits?
Sum Rule
Difference Rule
Constant Multiple Rule
Product Rule
Quotient Rule
Power Rule
Root Rule
Sum Rule
lim x→c (ƒ(x) + g(x)) = L + M
Difference Rule
lim x→c (ƒ(x) - g(x)) = L - M
Constant Multiple Rule
lim x→c (k × ƒ(x)) = k × L
Product Rule
lim x→c (ƒ(x) × g(x)) = L × M
Quotient Rule
lim x→c ƒ(x) / g(x) = L / M
Power Rule
lim x→c [ƒ(x)]ⁿ = Lⁿ, n a positive integer
Root Rule
lim x→c ⁿ√ƒ(x) = ⁿ√L = L^1/n, n a positive integer
How do you find a limit through algebraic manipulation?
by factoring the numerator and denominator, then canceling any removable discontinuities
When is finding a limit using algebraic manipulation most useful?
if you get limits where the denominator is equal to 0
Squeeze Theorem
Conditions:
For all values of x in the interval that contains a, g(x) ≤ f(x) ≤ h(x)
g and h have the same limit as x approaches a
lim g(x) = L, lim h(x) = L, therefore lim f(x) = L
What is the limit of [sin(x)/x] as x approaches 0?
1
What is the limit of [(cos(x)-1)/x] as x approaches 0?
0
What is the limit of [sin(ax)/x] as x approaches 0?
a
What is the limit of [sin(ax)/sin(bx)] as x approaches 0?
a/b
What is Jump Discontinuity?
occurs when the curve “breaks” at a particular place and starts somewhere else
the limits from the left and the right will both exist, but they will not match
Jump Discontinuity
What is Essential/Infinite Discontinuity?
the curve has a vertical asymptote
Essential/Infinite Discontinuity
What is Removable Discontinuity?
an otherwise continuous curve has a hole in it
“removable” because one can remove the discontinuity by filling the hole
Removable Discontinuity
What are the conditions for continuity?
For f(x) to be continuous when x=c:
f(c) exists
the limit as x→c exists
lim as x→c = f©
When is a function continuous?
if it is continuous at every point on that interval
How is it possible to remove discontinuities?
by redefining the function without that point in the domain
How is removing discontinuities frequently done?
by factoring out a common root between the numerator and denominator
What is a vertical asymptote?
a line that a function cannot cross because the function is undefined
What is a horizontal asymptote?
the end behavior of a function; it can be crossed
When is there no horizontal asymptote?
If the highest power x in rational expression is in the numerator and the limit as x approaches infinity is infinity
When is the horizontal asymptote y=0?
If the highest power of x is in the denominator and the limit as x approaches infinity is 0
When is the horizontal asymptote the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator?
If the highest power is the same in both the numerator and the denominator
What are the 2 ways of finding rate of change?
using the difference quotient to find the Average Rate of Change
finding it at a specific point in time, which is called the Instantaneous Rate of Change
What is the difference quotient?
the rate of change over an interval of time
Difference Quotient
What is the Instantaneous Rate of Change?
the difference quotient but with a limit as h → 0
How do you find the slope of a line that isn’t linear?
by using the secant line to approximate the slope
How do you find the slope of the secant line?
by using the difference quotient
How is the approximation of slope more accurate?
when the points are closer
What kind of line is more accurate to use when approximating slope of a curve and why?
the tangent line because it touches the curve at exactly one point
How can you get the tangent line?
by using the Instantaneous Rate of Change
Definition of Derivative
What is a derivative?
the rate of change at a specific point
What is the first and second derivative notation for f(x)?
f’(x) and f’’(x)
What is the first and second derivative notation of g(x)?
g’(x) and g”(x)
What is first and second derivative of y?
y’ or dy/dx and y’’
What are the derivative rules?
Constant Rule
Constant Multiple Rule
The Power Rule
The Product Rule
The Quotient Rule
What is the Constant Rule?
If f(c)=k where k is a constant then f’(x)=0
What is the Constant Multiple Rule?
If you have a constant multiplied by a function, you can “pull the constant out”
What is the Power Rule?
If f(x) = xⁿ then f’(x) = nx^n-1
A good way to describe this rule is to “multiply down and decrease the power”
The power rule works for polynomials
What is the Product Rule?
If f(x) = uv then f’(x) = (u)(dv/dx) + (v)(du/dx)
You take the first term and multiply it by the derivative of the second term then add that to the second term multiplied by the derivative of the first term
What is the Quotient Rule?
If f(x) = u/v then f’(x) = [(v)(du/dx) - u(dv/dx)]/v²
You take the denominator and multiply it by the derivative of the numerator, then subtract the numerator multiplied by the derivative of the denominator all over the denominator squared
Constant Rule
Power Rule
Product Rule
Quotient Rule
Chain Rule
What is the derivative of sin(x)?
cos(x)
What is the derivative of cos(x)?
-sin(x)
What is the derivative of tan(x)?
sec²(x)
What is the derivative of csc(x)?
(-csc(x))(cot(x))
What is the derivative of sec(x)?
(sec(x))(tan(x))
What is the derivative of cot(x)?
-csc²(x)
What is the derivative of ln(x)?
1/x
What is the derivative of e^x?
e^x
When do you use the chain rule?
if finding the derivative of a composite function
The Chain Rule
When do you take the derivative implicitly?
when you can’t isolate y in terms of x
What does implicit differentiation mean?
solving for the derivative of x with respect to y in order to get a derivative in terms of both variables
What is an easier way to describe implicit differentiation?
if your variable doesn’t match dx, then you need to follow it up with d(variable)/dx
Inverse Function Differentiation Formula
What does the derivative tell us?
the slope of the line tangent to the graph
What does the line tangent to the graph tell us?
the slope of the line at a particular point
What is position measured in?
meters
What is velocity measured in?
meters per second (m/s)
What is acceleration measured in?
meters/second²
What can we derive velocity to get?
the rate of change
What can we take the second derivative of position to get?
acceleration
What is the notation for position?
x(t) and sometimes s(t)
What is the notation for velocity?
x’(t) or v(t)
What is the notation for acceleration?
x’(t) or v’(t) or a(t)
When will particles speed up?
if the sign of velocity and acceleration match
(must be both either positive or negative)
What are related rates?
when change in one thing is related to change in another thing
How do you solve related rates problems?
read the problem carefully and identify all given information
draw a diagram if possible
determine what needs to be found and assign a variable to it
write an equation that relates the variables involved
differentiate both sides of the equation with respect to time
substitute in the given values and solve for the unknown rate
What should you do once you’ve solved a related rates problem?
ensure that units are included in our final answer and check that your answer makes sense in the context of the problem
What are differentials?
very small quantities that correspond to a change in a number
What is used to denote a differential?
∆x
What can differentials do?
approximate the value of a function
When is a limit called “indeterminate”?
if a limit gives you 0/0 or ∞/∞
What can you use to interpret an indeterminate limit?
L’Hospital’s Rule
L’Hospital’s Rule
Note 
Flashcard (21)