Limits
Name a type of discontinuity.
What is jump, infinite, removable, or endpoint
A function that models the slope of the curve for all x-values for which it exists. (definition)
What is the definition of the derivative?
The name for a and b on top and bottom of the integral symbol
What are Limits of integration?
Conditions for Rolle's Theorem.
1. f(x) is continuous on [a,b]
2. f(x) is differentiable on (a,b)
3. f(a) = f(b)
Process for finding horizontal asymptotes of a function.
What is finding the right-hand and left-hand limit as x approaches ∞ of the function?
Find the limit.
lim as x->2 of (x2-4)/ x2-x-2
4/3
Write the formula for the Product rule using f(x) and g(x).
Then write the formula for the Quotient rule. (use N for numerator and D for denominator)
Product rule: f'(x)g(x) + f(x)g'(x)
Quotient rule: [N'D - ND']/D2
Find the most general antiderivative of
f(x) = 4x2 + ex + 1/x
F(x) = 4/3x3 +ex + ln|x| + C
Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the following function:
f(x) = x3 + 2x2 - x on [-1, 2]
c = 0.7863
Use MVT: f'(c) = [f(b) - f(a)] / b - a
Find the average velocity of the particle over the time interval [2,6].
displacement = s = 2t2-3
avg. velocity = change in position/change in time
= s(6) - s(2)/6-2 = 16cm/s
Find the limit using L'hopital's Rule.
lim as x-->0 (5x3 + 2x) / sin x
L = 2
A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec at what rate is the radius decreasing when the area of the sheet is 12 m2?
(Use the equation for area of a circle: A = pi r2)
−0.040717 m/s
Integrate ∫x(2x+5)8dx
= (2x+5)10/40 - 5(2x+5)9/36 + C
Name the theorem.
If f is continuous on [a,b], then ∫(from a to b) f(x)dx = F(b) - F(a) where F is any antiderivative of f
(so F' = f)
What is the Fundamental Theorem of Calculus (part 2)
What is the natural rate of acceleration of a free-falling object in m/s^2?
-9.8m/s^2
Find the limit as h->0 of [√ (9+h) - 3]/h
L = 1/6
Find the velocity and acceleration given the position function.
s(t) = 2cos(t) + 3sin(t)
v(t) = -2sin(t) + 3cos(t)
a(t) = -2cos(t) - 3sin(t)
What is the difference between an indefinite and definite integral?
A definite integral has upper and lower limits on the integrals, and it's called definite because, at the end of the problem, we have a number – it is a definite answer. An indefinite integral is more of a general form of integration, and it can be interpreted as the anti-derivative of the considered function.
Is there a solution (a root) to f(x)= x5 - 2x3 - 2 = 0 between x=0 and x=2?
(What theorem do you use to know the answer)
Since this is a polynomial, it is continuous everywhere. f(0) = -2 and f(2) = 14 so according to the Intermediate Value Theorem, there must be some point between x=0 and x=2 that crosses the x axis where y=0, giving a solution. So Yes there is a solution.
A function fails to be differentiable if it has a....
What is a discontinuity, corner, cusp or a vertical tangent
Determine the infinite limit.
lim as x->5+ of (x+1)/(x-5)
L=∞
Find the equation of the tangent line of
f(x) = 3x3 - 6x2 + x at (1, 2).
y = -2x+4
Integrate ∫ 1/(x2 + 1) dx = ?
tan-1x + C
What does y= f''(x) mean and what is another notation for it?
This is the second derivative. Meaning the derivative of the derivative. Can be written as d2y/dx2 or as a(t) the acceleration function.
You can use the integral of the absolute value of velocity (aka integral of speed) to find what..?
What is distance traveled?