Calculus AB (Derivatives)
A function that is increasing and concave down has these properties.
What are f'>0 andf''<0?
\int_a^bf(x)dx=F(b)-F(a)
What is the fundamental theorem of calculus?
An improper integral is an infinite area. Thus the integral can only do one of these two things.
What is converge or diverge?
The general form of a logistic differential equation is given by this.
What is \frac{dP}{dt}=kP(1-\frac{P}{c}) ?
The sum of a convergent geometric series.
What is \frac{a}{1-r} ?
\frac{d}{dx}\cos^{-1}x=
What is -\frac{1}{\sqrt{1-x^2}}?
Finding a general solution to a differential equation requires this technique.
What is separation of variables?
The product rule for integration is known as this.
What is integration by parts?
When using Euler's method, the value
Delta x
is also known as this.
What is the step size?
A sequence {a_n} whose terms can be described as a_{n+1}\leq a_n for all n is called this.
What is nonincreasing?
A function is differentiable if and only if it is this.
What is continuous?
\frac{d}{dx}\int_0^{x^2}\sqrt{t^2-1}dt=
What is
2x\sqrt{x^4-1}?
When using integration by parts, the choice for u can be determined by using an acronym that stands for this.
What is Logarithmic, Inverse Trig, Algebraic, Trig, Exponential?
A logistic population function is at maximum growth when the population reaches this
What is half the carrying capacity?
When a series involves factorials, it is usually best to use this test to determine convergence.
What is the ratio test?
A derivative is defined by this limit.
What is \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}?
If \int_0^1f(x)dx=2 and \int_0^2f(x)dx=4 then \int_0^1f(2x)dx=
What is 2?
An integrand that is the product of an algebraic terms and either an exponential or a trigonometric terms can use this method to integrate quicker.
What is the tabular method?
Euler's method is this kind of process.
What is iterative?
For convergence, an alternating series must meet these conditions.
What are decreasing and
\lim_{n\rightarrow\infty}a_n=0?
In a related rate problem, the necessary equation is differentiated with respect to this quantity.
What is time?
The equation \int_{x_0}^{x_1}f'(x)dx calculate this.
What is the net change in
f?
The method of partial fraction decomposition allows a rational expression to be rewritten in terms of a sum of fractions whose denominators are these of the original denominator.
What are linear factors?
If a population is modeled by the differential equation \frac{dP}{dt}=0.01P(100-P) then \lim_{t\rightarrow\infty}P(t) =
What is 100?
\frac{e}{pi}+\frac{e^2}{\pi^2}+\frac{e^3}{\pi^3}+\cdots=
What is
\frac{e}{\pi-e}?