Theorems
If a function f is continuous on [a, b] and differentiable on (a, b), this theorem guarantees at least one value c in (a, b) where the instantaneous rate of change equals the average rate of change.
What is the Mean Value Theorem (MVT)?
d/dx (f(g(x)) = f'(g(x)) * g'(x)
What is the Chain Rule?
This integration technique, often remembered by the acronym LIATE, is based on the product rule for differentiation and uses the formula int u * dv = uv - int v * du.
What is Integration by Parts?
This is the standard algebraic technique used to solve basic differential equations like dy/dx = x/y by moving all x terms to one side and all y terms to the other.
What is Separation of Variables?
An infinite series of the form 1/n^p converges if and only if this condition is met by the exponent p.
what is p > 1?
This theorem states that if f is continuous on a closed interval [a, b], then f must attain both an absolute maximum and an absolute minimum value on that interval.
What is the Extreme Value Theorem (EVT)?
To find the slope of a curve defined parametrically by x = f(t) and y = g(t), you evaluate this ratio of derivatives.
what is dy/dt / dx/dt or g'(t) / f'(t)?
This algebraic technique is used to evaluate integrals like int 1/x^2 - x * dx by breaking the integrand down into a sum of simpler fractions, A/x + B/x-1.
What is Partial Fraction Decomposition?
This is a visual representation of a differential equation, consisting of short line segments representing the slope at various coordinates (x, y).
What is a Slope Field?
This test states that if the limit of the terms of a series as n approaches infinity is not equal to zero, the series must diverge.
What is the nth Term Test for Divergence?
If f is continuous on [a, b] and d is a number strictly between f(a) and f(b), this theorem guarantees there is at least one value c in (a, b) such that f(c) = d.
What is the Intermediate Value Theorem (IVT)?
This is the derivative of the polar function r = f(x) with respect to x for the curve r = 3 - 2sin(x) at x = pi.
What is 2?
An integral is classified as this specific type if it has an infinite limit of integration or an infinite discontinuity within the interval of integration.
What is an Improper Integral?
This numerical method uses a step size delta x to approximate a solution to a differential equation by iteratively calculating y new = y old + dy/dx * change in x.
What is Euler’s Method?
This is the specific name given to a Taylor series that is centered at x = 0.
What is a Maclaurin Series?
This specific part of a major calculus theorem states that if f is continuous on [a, b] and F(x) = integral from 0 to x of f(t) dt and F'(x) = f(x).
What is the First Part of the Fundamental Theorem of Calculus?
If a function f(x) has an inverse function g(x), and f(3) = 5 with f'(3) = 2, this is the value of g'(5).
what is 1/2?
This is the integral expression used to find the exact arc length of a smooth curve y = f(x) from x = a to x = b.
What is int from a to b of (1+[f'(x)]^2)^1/2 * dx?
The differential equation dP/dt = kP(1 - P/M) models this type of bounded population growth, where M represents the carrying capacity.
What is Logistic Growth?
By using the Ratio Test on a power series, you can find the set of all x-values for which the series converges, which is known by this term.
What is the Interval of Convergence?
This theorem provides an upper bound for the error when approximating a function using a polynomial, stating that the error Rn(x) is equal to M/(n+1)! * (x-a)^n+1 for some c between and x. M represents the maximum value of n+1 derivative.
What is the Lagrange Error Bound?
This indeterminate form limits-solving technique allows you to differentiate the numerator and denominator separately when encountering 0/0 or infinity/infinity.
What is L'Hôpital's Rule?
To find the area of the region enclosed by the polar curve r = f(theta) from theta = a to theta = b, you evaluate this definite integral.
What is 1/2 int from a to b of r^2 * dtheta?
For the logistic growth model dP/dt = 0.2P(1 - P/800), this is the population size P at which the population is growing the fastest.
What is 400?
This is the full Maclaurin series expansion for the function f(x) = e^x.
What is summation from n = 0 to infinity of x^n / n! or 1 + x + x^2/2! + x^3/3! + x^4/4!?