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[image]
Noah's mnemonic: Lodeehi-Hideelo |
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| Where can the graph f(x) have an inflection point? |
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when f''(x) = 0
when the graph of the second derivative intersects the x-axis |
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| Where can the graph f(x) have extrema? |
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when f'(x) = 0
when the graph of the first derivative intersects the x-axis |
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[image]
Adding more terms increases the accuracy of the function. |
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| same as Taylor series, except [image] |
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| formulas for Newton's Law of Cooling |
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[image]
If [image], then [image]
If [image], then [image]
T = temperature of object at time t
TA = temperature of room
t = amount of time that has passed
c, k = constants to be found
Go here and here for more detailed explanations. |
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For a continuous, differentiable function f(x) bounded within [a,b]...
If you draw a line from (a, f(a)) and (b, f(b)), there is at least one tangent line of f(x) within a ≤ x ≤ b parallel to that line. [image] |
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a special case of the Mean-Value Theorem
If f(a) = f(b), then there is at least one critical point within a ≤ x ≤ b.
[image] |
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| Intermediate-Value Theorem |
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If two points f(a) and f(b) are connected by a continuous curve f(x), and there is a line above f(a) but below f(b), f(x) must cross the line at some point!
[image] |
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| What does it mean when f'(x) > 0? |
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| What does it mean when f'(x) < 0? |
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| What does it mean when f''(x)>0? |
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| What does it mean when f''(x)<0? |
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| Finding horizontal asymptotes of rational expressions |
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Find the degrees of the numerator and denominator, n and d.
If n < d, then the asymptote is y = 0.
If n = d, then the asymptote is y = c, where c is the quotient of the leading coefficients of the top and bottom polynomials.
If n = d+1, see "Oblique asympotes."
If n > d+1, then there are no horizontal or oblique asymptotes.
Note: If [image], then y = b is a horizontal asymptote of f(x). |
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| Finding vertical asymptotes of rational expressions |
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Find the zeroes of the denominator.
Note: If [image], then x = a is a vertical asymptote of f(x). |
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| Finding oblique (diagonal) asymptotes of rational expressions |
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Find the degrees of the numerator and denominator n and d.
If n=d+1, then do polynomial division. The asymptote is y=c, where c is the quotient with no remainder. |
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If f(a) = g(a) = 0 or ∞, then [image]
*The limit has to exist for the rule to apply.
L'Hopital's rule can be applied over and over until the limit is no longer indeterminate. |
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| Fundamental Theorem of Calculus |
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1. [image], where F'(x) = f(x)
2. If [image], then G'(x)=f(t)
In other words, the integrand is the derivative of the integral |
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| The average of all values of f(x) in the interval [a,b] |
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| Formula for average rate of change |
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[image] or [image]
For speed, find the absolute value of the integrand |
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| Formula for instantaneous rate of change |
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f'(x) or [image] at x = a
For speed, find the absolute value of the derivative |
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| Formula for greatest speed |
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f'(a), where f''(a) = 0
Find the absolute value of this derivative |
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| Displacement vs. distance traveled |
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| Displacement is the final position of an object (can be positive or negative). Distance traveled is the length the object moved to get to its final position (has to be positive). |
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Cartesian: [image]
Parametric: [image]
Polar: [image] |
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Cartesian: [image]
Parametric: [image]
Polar: [image] |
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| Rotations about an axis: disks |
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[image] if rotated around the x-axis (r=y(x))
[image] if rotated around the y-axis (r=x(y)) |
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| Rotations about an axis: washers |
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[image] if rotated around the x-axis (r=y(x))
[image] if rotated around the y-axis (r=x(y)) |
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Rotations about an axis: shells
(or Mr. Cocharo's toilet paper rolls) |
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[image] if rotated about the x-axis (r=y and w=width=x(y))
[image] if rotated about the y-axis (r=x and h=height=y(x)) |
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| Sandwich or Squeeze Theorem |
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| If f(x) ≤ g(x) ≤ h(x), and if [image], then [image]. [image] |
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| What about rotations about a line (x ≠ 0 or y ≠ 0) instead of an axis? |
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For line x = a,
For the disk method, r = a-y(x)
For the ring method, R = a-y(x) and r = a-y(x)
For the shell method, h = a-x(y)
Do the same for x(y) if the line is y = a.
Go here and here for good examples of what I'm talking about. |
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| Hyperbolic trigonometric identities to know (2) |
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| Trigonometric identities to know (9) |
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A. [image]
B. [image]
C. [image]
D. [image]
E. [image]
F. [image]
G. [image]
H. [image]
You can use A to modify H if needed for a problem.
I. [image] |
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| How to find: displacement and distance traveled |
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| Use the formulas for bounded area. For distance traveled, find the absolute value of the integrand. |
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| How to find: derivative of a parametric curve |
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| How to find: derivative of a polar curve |
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First parametrize the polar equation!
r(θ) → x = r×cos(θ) and y = r×sin(θ)
[image] or [image] |
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| How to do: tangent-line approximation for f(x) at x = a |
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1. Find f(x) and f'(x) at x = b, where b is a rounded to the nearest integer.
2. The tangent line equation is y = f(b)+f'(b)×(x-b).
3. Plug in |a-b| for x to solve for y. |
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the position of a point P on a parametrically defined curve
[image] |
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| the velocity of a point P on a parametrically defined curve
[image]
The magnitude of this vector is the speed of a particle at point P. |
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| Define: acceleration vector |
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the acceleration of a point P on a parametrically defined curve
[image] |
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| Things to know: integration by partial fractions |
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| For some cards, I can't give a concise definition or how-to. So instead, I'll just remind you that this is something you need to know. |
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| Things to know: find the dimensions for the smallest surface area given a volume |
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| For some cards, I can't give a concise definition or how-to. So instead, I'll just remind you that this is something you need to know. |
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| Things to know: find the area of the largest polygon that fits inside another polygon or a curve |
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| For some cards, I can't give a concise definition or how-to. So instead, I'll just remind you that this is something you need to know. |
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| Maclaurin series for sin(x) |
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| Maclaurin series for cos(x) |
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| Maclaurin series for 1/1-x |
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| Maclaurin series for 1/1+x |
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| Maclaurin series for ln(1+x) |
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| Maclaurin series for arctan(x) |
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| Domain and range of sin(x) |
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Domain: all real values of x
Range: -1 ≤ y ≤ 1 |
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| Domain and range of cos(x) |
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Domain: all real values of x
Range: -1 ≤ y ≤ 1 |
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| Domain and range of tan(x) |
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Domain: all real values of x, except x = π/2+nx
Range: all real values of y |
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| Domain and range of arcsin(x) |
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Domain: -1 ≤ x ≤ 1
Range: -π/2 ≤ y ≤ π/2 |
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| Domain and range of arccos(x) |
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Domain: -1 ≤ x ≤ 1
Range: 0 ≤ y ≤ π |
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| Domain and range of arctan(x) |
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Domain: all real values of x
Range: -π/2 ≤ y ≤ π/2 |
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Domain: all real values of x
Range: y ≥ 0 |
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| Domain and range of log(x) |
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Domain: x > 0
Range: all real values of y |
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| Domain and range of ln(x) |
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Domain: x > 0
Range: all real values of y |
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Domain: all real values of x
Range: y > 0 |
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| Given f'(t) and f(0), find f(t). |
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| When is the speed of a particle increasing? |
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| when v(t) and a(t) have the same signs |
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| When is the speed of a particle decreasing? |
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| when v(t) and a(t) have different signs |
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| When does a particle change direction? |
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| If f(x) and f^-1(x) are inverse functions, then |
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| If [image] converges, then [image]. |
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| If [image], then [image] diverges. |
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A geometric series [image]converges if and only if |r| < 1.
If |r| < 1, the sum is [image]. |
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| If f(x) is a continuous, positive, decreasing function and f(n) = a(n), then [image] converges if and only if [image] converges. |
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| A p-series [image] converges if p > 1 and diverges if p < 1. |
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| If [image], then ∑a(n) converges if L < 1 and diverges if L > 1. |
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| If [image], then ∑a(n) converges if L < 1 and diverges if L > 1. |
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For two positive series [image] and [image], where a(n) ≤ b(n)...
1. If ∑b(n) converges, then so does ∑a(n).
2. If ∑a(n) diverges, then so does ∑b(n).
Note: ∑a(n) and ∑b(n) have to start at the same n-value. |
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| If [image] is finite and nonzero, then ∑a(n) and ∑b(n) both converge or both diverge. |
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An alternating series converges if
1. a(n+1) < a(n) for all n-values
2. [image] |
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