Limits
f(x)= (x-2) / (x^2-4)
What is the removable and infinite discontinuity?
x=2 is removable
x=-2 is infinite
If f is continuous on the closed interval [a, b], f(a)≠f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k.
What theorem is this?
What is the intermediate value theorem?
Find f'(x).
f(x) = x^4 + 5x^3 - 7x + 2
What is f'(x) = 4x^3 + 15x^2 - 7
When does a particle speed up?
What is velocity and acceleration are both the same sign.
Evaluate.
2∫5 3x-5 dx
What is 33/2
What is the vertical asymptote?
g(x)=(3x+15) / (x^2+7x+10)
What is x= -2
Which test uses the critical value of a continuous and differentiable function on an open interval to find the relative extrema using the direction of slope.
What is the first derivative test.
Find f'''(x)
f'(x) = (7x - 3)^2 - ln 3x
*Calculator permitted*
What is 98 - (1 / x^2)
What does the 0∫5 v'(t) dt represent?
What is the change in velocity from t = 0 to t = 5.
Is a trapezoid estimation concave down an overestimation or underestimation?
What is underestimation
What is the horizontal asymptote?
lim h(x) (e^3) / (x^2)
x-->infinity
What is 0
f(x)is continuous on the closed interval [a, b], and f(x)is differentiable on the open interval (a, b). Then, there is a number, c, such that a < c < b and f'(c) = (f(b)−f(a)) / (b-a).
What theorem is this?
What is the Mean Value Theorem
Find anti derivative.
f(x) = 3x^2 - sec^2(x) + 5x - 9
What is x^3 - tan(x) + (5x^2 / 2) - 9x + C
When is f(x) concave up?
What is when f'(x) is increasing and when f''(x) is positive.
Given f(x) = 2x^2 + 5x - 2
Use right-riemann sum on the interval [-1,3] for four sub intervals.
*Calculator permitted*
What is 50.
Evaluate
lim (e^2x - 1) / x
x->0
What is 2
If f is continuous on the interval [a,b], then the area under the curve of f from [a,b] can be represented by
a∫b f(x) dx = F(x) a|b = F(b) - F(a)
What theorem is this?
What is the Fundamental Theorem of Calculus
What is the average value of g(x) on the interval [-2,6]
g(x) = (sin x) / 8
What is (cos(2) - cos(6)) / 64
Given function h(x) = 4x^2 - 8
Which intervals does h(x) attain a local minimum?
*Calculator permitted*
What is x = 0
Given g(x) = sin(pi/x)
f(x) = 4x - (1/x)
Find h(x) = 5f(g(4)) - f(1)
What is (5sqrt2) - 3
f(x) = { (x^2) x < 3
(5x-6) 3 ≤ x ≤ 5
(ln 6 + x^2) x >5
lim f(x)
x->3
What is 9
A continuous function f(x)on a closed interval [a, b] attains both an absolute maximum for all x and an absolute minimum for all x in the interval.
What theorem is this?
What is the Extreme Value Theorem.
f(x) = (9x^2 - sinx) / (4x-1)
What is (36x^2 - 18x - 4xcos(x) + cos(x) + 4sinx) / (4x-1)^2
What conditions must be met to find point of inflection for the second derivative?
What is f'''(x) has to have a critical point and is changing sign.
If dy/dx = -5ycos(x), find the particular solution of y when the initial condition is y(0) = 1
What is y(x) = e^(-5sinx)
C = 1