CALCULUS FLASH CARDS
Terms in this set (125)
Graph of y = sin x
Graph of y = cos x
Graph of y = tan x
Graph of y = sec x
Graph of y = 1/x
Graph of y = √x
Graph of y = √(1 - x^2)
Graph of y = |x|
Graph of y = ln x
Graph of y = e^x
An even function is ...
Symmetric with respect to the y-axis
i.e. y = x^2, y = cos x, y = |x|
f(x) = f(-x)
An odd function is ...
Symmetric with respect to the origin
i.e. y = x^3, y = sin x, y = tan x
f(x) = -f(x)
Two formulas for the area of a triangle
A = (1/2)bh
A = (1/2)absinC
Formula for the area of a circle
Formula for the circumference of a circle
Formula for the volume of a cylinder
Formula for a volume of a cone
Formula for the volume of a sphere
Formula for the surface area of a sphere
Point-slope form of a linear equation
Limit of (sin x) / x as x → 0
(lim x→0 (sin x / x))
Limit of (cos x) / x as x → 0
(lim x→0 (cos x / x))
A tangent line is ...
The lie through a point on a curve with slope equal to the slope of the curve at that point
A secant line is ...
The line connecting two points on a curve
A normal line is ...
The line perpendicular to the tangent line at the point of tangency
f(x) is continuous at x = c when ...
1. f(c) exists
2. limit of f(x) approaching c exists
3. limit of f(x) approaching c is equal to f(c)
f ' (x) =
f ' (h) =
What f ' (x) tells you about a function
1. Slope of curve at a point
2. Slope of tangent line
3. Instantaneous rate of change
Average rate of change is ...
Power rule for derivatives
Product rule for derivatives
Quotient rule for derivatives
Chain rule for derivatives
Derivative of sin x =
Derivative of cos x =
Derivative of tan x =
Derivative of cot x =
Derivative of sec x =
sec x tan x
Derivative of csc x =
-csc x cot x
Derivative of arcsin x =
Derivative of arccos x =
Derivative of arctan x =
Derivative of arccot x =
Derivative of arcsec x =
Derivative of arccsc x =
Derivative of ln x =
Derivative of log base a =
Derivative of e^x =
Derivative of a^x =
a^x ln a
Derivative of inverse function =
Rolle's Theorem: if f is continuous on [a, b], differential on (a, b), and f(a) = f(b), then ...
There exists c such that f ' (c) = 0
Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), then ...
There exists c such that ...
Extreme Value Theorem: if f is continuous on a closed interval, then ...
f must have both an absolute maximum and absolute minimum on the interval
Intermediate Value Theorem: if f is continuous on [a, b], then ...
f must take on every y-value between f(a) and f(b)
If a function is differentiable at a point, then ...
It must be continuous at that point
Four ways in which a function can fail to be differentiable at a point:
4. Vertical tangent line
A critical point of f(x) is ...
A value of x where f ' (x) = 0 or does not exist
If f ' (x) > 0, then ...
f(x) is increasing
If f ' (x) < 0, then ...
f(x) is decreasing
If f ' (x) = 0, then ...
f(x) has horizontal tangent line
f(x) is concave up when ...
f ' (x) is increasing
f(x) is concave down when ...
f ' (x) is decreasing
f '' (x) > 0 means that f(x) is ...
Concave up (like a cup)
f '' (x) < 0 means that f(x) is ...
Concave down (like a frown)
An inflection point is where ...
To find an inflection point ...
Look for where f '' (x) changes signs or where f ' (x) changes direction
To find extreme values of a function, look for where ...
f ' (x) is zero or undefined (critical numbers)
At a maximum, the value of a derivative is ...
f ' (x) changes from positive to negative
At a minimum, the value of a derivative is ...
f ' (x) changes from negative to positive
The Second Derivative Test: if f ' (x) = 0 ...
And f '' (x) < 0 then f has a maximum
And f '' (x) > 0 then f has a minimum
Position function: s(t) =
Antiderivative of velocity
Veolocity function: v(t) =
Derivative of position
Antiderivative of acceleration
Acceleration function: a(t) =
Derivative of velocity
Second derivative of position
A particle is moving left when ...
v(t) < 0
A particle is moving right when ...
v(t) > 0
A particle is a rest when ...
v(t) = 0
A particle changes direction when ...
v(t) changes signs
To find displacement of a particle with velocity v(t) from t = a to t = b, calculate the:
To find total distance traveled by a particle with velocity v(t) from t = a to t = b, calculate the:
Area between curves
Integral of top function minus bottom function
Volume of a solid with cross-sections of a specific shape
Integral of area of cross-section
Volume using discs
Volume using washers
Volume using shells
Area of a trapezoid
Trapezoidal rule for the derivative of f(x)
Average value of f(x) on [a, b]
Power rule for antiderivatives
Constant multiple rule for antiderivatives
Antiderivative of sin x =
-cos x + C
Antiderivative of cos x =
sin x + C
Antiderivative of sec^2 x =
tan x + C
Antiderivative of csc^2 x =
-cot x + C
Antiderivative of sec x tan x =
sec x + C
Antiderivative of csc x cot x =
-csc x + C
Antiderivative of 1/x =
ln |x| + C
Antiderivative of e^x =
Antiderivative of 1/√(1-x^2) =
arcsin x + C
Antiderivative of 1/(1+x^2) =
arctan x + C
L'Hopital's rule for indeterminate limits
Fundamental Theorem of Calculus Part 1:
Fundamental Theorem of Calculus Part 2:
A differential equation is ...
An equation containing one or more derivatives
To solve a differential equation ...
Separate the variables
Integrate both sides
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