Calculus AB & BC

Points where the derivative is 0

Points where the derivative is undefined

When the velocity is 0.

Plug t value back into the original equation to find the points.

They are constants that can be pulled out from both equations

Logs, Inverse, Algebraic, Trig, Exponential

When the bounds are negative or positive infinity, or it has a discontinuity

Express fraction as a sum of polynomial and several fractions with a simpler denominator

Terms listed by commas related to each other often through a pattern/rule

Sum of the terms of a sequence (converge or diverge)

If the limit of the series equals anything other than 0, including positive or negative infinity, then the series diverges. If the limit of the series equals 0, then the Nth Term Test is inconclusive.

Terms cancel out with one another in a certain way.

Find out if series converges by taking the improper integral of a series

An infinite series where terms are 1/integer to the p power. It converges if p>1, and it diverges if 0< p </= 1.

If the greater series converges, so does the smaller one. If the smaller series diverges, so does the greater series . If the smaller series converges, it is inconclusive by the direct comparison test.

If you have two series, and they are both greater than 0, and the limit of an/bn = L, then they either both converge or diverge.

A series that alternates from positive to negative, if the limit of an (the part that doesn’t alternate) as n approaches infinity is 0, and a_n is a decreasing sequence, then the series converges.

This approximates the sum of the alternating series that converges. Practice how to approximate the interval of the sum.

Degree of the denominator is greater than the degree of the numerator, horizontal asymptote when y = 0.

The degree of the numerator and the denominator are the same, and the horizontal asymptote is the ratio of the leading coefficients.

The degree of the numerator is greater then the degree of the denominator. No horizontal asymptote, there is a slant asymptote instead.

If absolute value of a_n converges, then the entire series converges absolutely.

If the absolute value of a_n doesn’t converge, then the entire series converges conditionally. You can take the derivative of the series, and if it is negative, then the function is decreasing.

A sequence with a constant ratio for each term (multiplied, exponential, or divided by)

A sequence where the difference between each consecutive term is constant.

For example, the rate of population growth as it tapers off. L, the limiting value, is the value (asymptote) the line eventually reaches. The inflection point is the point where the line changes from concave up to concave down.

If a function has derivatives at all orders of x = c, then the series is called the Taylor series for f at c.

Special Type of Taylor Series where c = 0.

A finite number of terms from the Taylor Series (ex. a taylor polynomial that has a degree of 3, would have the first 3 terms: n = 0, n = 1, n= 2, and n = 3).

How close a remainder is to the actual value shows how good the approximation is. The exact value (f(x)) is equal to the approximate value (P(x)) plus the remainder (R(x)).

factorial>exponential>power functions> log functions

Use ratio test to find interval and radius of convergence of a power series. If L is greater than 1, series is divergent. If L is less than 1, the series is absolutely convergent. If L = 1, the test is inconclusive. Test endpoints by plugging in values for x and seeing if it converges and diverges (need to conduct another series test to do this).

No need to test endpoints if asking for ROC. Find distance (left and right) from c value of function where series converges. If the problem says “must be true”, then it can be proven without a calculation (watch wording).

Series where function is to the nth power, centered around some c value. Can be multiplied by coefficient of (a).

If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point in (a,b) then there is at least one point “c” in (a,b) at which f’(c ) = f(b) - f(a) / b - a.

If the graph of f(x) is concave up, the rate of change is increasing. If the graph of f(x) is concave up, f’(x) is increasing, and f’’(x) is positive.

If the graph of f(x) is concave down, the rate of change is decreasing. If the graph of f(x) is concave down, f’(x) is decreasing, and f’’(x) is negative.

If a function f has a local max or min at an interior point “c” of its domain, and if f’(x) exists at c, then f’(c ) = 0. Remember, f’(c ) might not exist.

• Points where f’(x) = 0

• Points where f(x) does not exist

• End points of the domain interval

If f is continuous on a closed interval [a,b] then f has both a maximum value and a minimum value on the interval. The maximum value is the absolute max, and the minimum value is the absolute minimum. We use the extreme value theorem to see if there is a guarantee that there is an absolute maximum and minimum. If there is no guarantee, there might still be a max or min. If the function isn’t defined on a closed interval, there is no guarantee. (Note: the function doesn’t need to be explicitly defined to be on a closed interval, some functions have restricted domains.)

NOT FINISHED

• Separate dy and dx

• Take the integral of both sides

• Isolate until y = …

Integration Shortcut

• y = |x| at x = 0 (corner discontinuity)

• y = x^(2/3) at x = 0 (cusp discontinuity)

• y = ³√x at x = 0 (vertical tangent)

The left hand derivative and right hand derivative are not equal, so the derivative does not exist.

One side derivative goes to infinity, and the other side derivative goes to negative infinity. So, dy/dx is approaching infinity or negative infinity.

Both sides approach infinity or negative infinity.

This function does not have a derivative because it is discontinuous at that x value.

Average value of a function over an interval [a,b]. This means the average of all the y values of the function over the interval [a,b].

Shortcuts for solving definite integrals

Given a displacement time graph, the slope of the tangent line provides the velocity at a point. We can find displacement by doing area under time. Area under curve and slope of the tangent should be connected.

Implicit Differentiation is the process of finding dy/dx for such functions. The chain rule plays an important part in this differentiation.