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2.1: Preview of Calculus
2.2: Limit of a Function
2.3: The Limit Laws
2.4: Continuity
2.5: Epsilon-Delta Proof
100
The slope of a line tangent at a given point on a curve. The _________ rate of ________?
What is instantaneous rate of change?
100
Read aloud: lim x --> 2 f(x) = 4
What is the limit as x approaches two of f of x is 4?
100
The limit as x approaches 3 of 2x - 4.
What is 2?
100
A hole is an example of this type of discontinuity.
What is a removable discontinuity?
100
Epsilon and delta both represent this in calculus proofs.
What is a small distance?
200
Consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. It's height in meters at time t seconds is h(t) = 15t - 4.9t^2. The average rate of change over the time interval from [1, 1.05].
What is 206.995 m/s?
200
A limit approaching a value with a superscripted +, is a(n) _____-sided limit approaching from the ________.
What is one-sided limit approaching from the right?
200
The limit as x approaches pi of -sec(2x).
What is -1?
200
Determine whether f(x) = (x+2)/(x+1) is continues at the point x = 1. If not, name the type of discontinuity.
What is an infinite discontinuity? (Essential)
200
The epsilon delta definition for lim x --> a (f(x)) = N.
What is for every epsilon greater than zero, there exists a delta greater than zero, so that if zero is less than x - c is less than delta, then |f(x) - N| < epsilon?
300
Given the function f(x) = cos x and a point on the graph P(1.5, 0), use a table of values to find the slope of the line tangent to the function at x = 1.5.
What is m = pi?
300
When a limit approaches infinity or negative infinity, use this principle to determine the limit.
What is the principle of dominance?
300
The limit as x approaches -3 of (x+3)/(x^2 + 4x + 3).
What is -1/2?
300
The interval over which f(x) = sqrt (4-x^2) is continuous.
What is [-2, 2]?
300
For the lim x--> 4 of 2x = 8, find the smallest delta given epsilon = 0.1.
What is 0.05?
400
Given the function f(x) = x^2 - 4, use a table of values to find the slope of the line tangent to the function at x = 1.
What is m = 2?
400
lim x--> 0^- (||x||)
What is -1?
400
The limit as x approaches infinity of (x+9)/(x^2 + 9).
What is zero?
400
The rule that states that for a continuous function over a closed, bounded interval [a, b], if there is any real number z between f(a) and f(b), then there is a number number c in [a,b] satisfying f(c) = z.
What is the Intermediate Value Theorem?
400
For the lim x--> 4 of (1/5)x = 4/5, find the smallest delta given epsilon = 0.01.
What is 0.05?
500
Given the function f(x) = 2x^3 + 1, use a table of values to find the equation of a line tangent to the graph at x = 2.
What is y = 24x - 31?
500
If lim x--> 3^- = 1, and lim x-->3^+ = infinity, then the limit as x approaches 3 is this.
What is does not exist?
500
The limit as x approaches zero of (sqrt (1+x) -1)/x.
What is 1/2?
500
Use the IVT to determine whether 2^x = x^3 has a solution in one of the intervals [1.25, 1.375] or [1.375, 1.5].
What is [1.25, 1.375]?
500
Give an epsilon delta proof for the limit as x approaches 3 of 3x + 1 = 10.
What is see Mrs. Kuiper?