Limits and Continuity
The value of the limit as x approaches 0 of sin(x)/x
1
d/dx of ln(x)
1/x
A point on a graph where the concavity changes
inflection point
The indefinite integral of 1/x with respect to x
ln|x|+c
If f'(x) > 0 and f''(x) < 0, the graph of f(x) is described by these two visual properties.
increasing and concave down
The type of discontinuity found at x = 2 for the function f(x) = x-2/x^2-4
removable discontinuity
This rule is used to find the derivative of a composite function like f(g(x))
chain rule
The rate of change of velocity with respect to time
acceleration
The geometric interpretation of a definite integral
area under a curve
W(t) represents the tons of waste in a landfill, this is the real world meaning of W'(5) = 2.
the amount of waste is increasing at a rate of 2 tons per year at year 5
This theorem guarantees that if f(x) is continuous on [a, b], it takes on every value between f(a) and f(b).
intermediate value theorem or IVT
d/dx of tan(x)
sec^2(x)
This theorem states that if a function is continuous on [a,b] and differentiable on (a,b), there is a point c where the instantaneous rate of change equals the average rate of change.
mean value theorem or MVT
This method of integration reverses the chain rule
u substitution
A local minimum occurs on the graph of f(x) when the graph of f'(x) makes this specific transition.
changing from negative to positive
The horizontal asymptote of the function f(x) =3x^2 - 5/2x^2 + 7
y=3/2
The calculus rule used to evaluate limits that result in indeterminate forms like 0/0 or infty/infty
Lhopitals rule
If a radius of a circle is expanding at 2 cm/s, this is the rate at which the area is changing when the radius is 5 cm.
20pie cm^2/s
The derivative with respect to x of x^2to\1 cos(t)dt
2xcos(x^2)
C(t) is the temperature of coffee in degrees Celsius, this is the units of measurement for C''(t) if time is in minutes.
degrees celsius per minute squared
The value of the limit as x approaches infinity of (1+1/x)^x
e
d/dx of 5^x
5^xln(5)
According to the Extreme Value Theorem, these are the two types of locations where absolute extrema can occur on a closed interval.
critical points and endpoints
This approximation method uses the average of the Left Riemann Sum and Right Riemann Sum.
trapezoidal rule
For a differentiable function f(x), if f'(3) = 0 and f''(3) = -4, this is the behavior of the graph at x = 3 according to the Second Derivative Test.
local maximum