Limits and Continuity
A requirement for a limit to exist.
What is the one-sided limits are equal?
\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
What is
f'(x)?
In a related rate problem, the equation of interest is differentiated with respect to this quantity.
What is time?
An antiderivative has the form \int f(x)dx=F(x)+C where C is called this.
What is the constant of integration?
There are this many defined points on the unit circle.
What is 16?
When a function approaches a finite value as x approaches infinity, then the function has one of these.
What is a horizontal asymptote?
A function that is increasing and concave up has both of these properties.
What is
f'(x)>0 and f''(x)>0?
Optimization problems involve these two type of equations.
What are objectives and constraints?
Geometrically, an integral of a function calculates this value from x=a to x=b .
What is the area under the curve?
If f(x)=\ln x then f(0) is undefined. However, as x approaches 0 from the righthand side, f(x) approaches this.
What is negative infinity?
When a function has a removable discontinuity at x = a, the graph of the function has this feature at x = a.
What is a hole?
A function is differentiable if and only if it is this.
What is continuous?
The Mean Value Theorem applies to all functions that are this.
What is continous and differentiable?
The chain rule version of integration is known as this.
What is u-substitution?
The tangent function is undefined at odd multiples of this angle.
What is
\frac{\pi}{2}?
When a function is continuous at x = a, then
\lim_{x\rightarrow a}f(x) is equal to this.
What is
f(a)?
What are extrema?
L'Hopital's Rule can be applied to limits involving these indeterminate forms.
What are
\frac{0}{0} and \frac{\infty}{\infty}?
Moving from left to right, a function whose area is underneath the x-axis is this.
What is negative?
The sine and cosine of an angle are equal to each other at every other odd multiple of this angle.
What is
\frac{\pi}{4}?
\lim_{x\rightarrow -\infty}e^x=
What is zero?
A function is not differentiable at points that have one of these three features.
What are cusps, vertical tangents and discontinuities?
A sphere of radius r has a volume V=\frac{4}{3}\pi r^3 . If the radius is changing with time, then the rate of change of the volume with respect to time is given by this.
What is
V'=4\pi r^2r'?
If g(x)=\int_0^xf(t)dt , then the derivative of g is this.
What is
f(x)?
Calculus makes people do this.
What is cry?