Anything from Units #1 - #3 and
Unit #6
Unit #6
Please determine if the following piecewise function is continuous at x = 0 and provide an oral justification.
NOTE: No points will be earned with the lack of Justification AND No points will be earned for incorrect answer
f(x) = (eX) - e + 8 for 0 < x < 1
= 3e(X-1) + 4 for 1< x <3
This function is NOT Continuous.
The limit as x approaches one from the left (8) is not equal to the limit as x approaches one from the right (7).
If f(x) is a cts and differentiable function and f'(c) are either equal to 0 or DNE... Which of the following is the only thing you can conclude about the point x = c on f(x)
a) f(x) has a local/relative minimum or maximum at x = c
b) f(c) is a critical point to f(x)
c) f(c) is a point of inflection
d) f(c) does not exist
NOTE: Double points can be earned if the answer provided explains why one of the answers not chosen is incorrect.
NOTE #2: If the incorrect answer is chosen, but the answer choice provided for the double point opportunity is wrong, AND the justification is correct, 100 points may be earned by the given team
NOTE #3: If the team gives the incorrect answer AND does not earn the double points, the team will lose 300 points
B) --> f(c) is a critical point to f(x)
A is incorrect because we must use the first derivative test to determine if f'(x) changes the sign from either negative to positive or vice versa to determine if f(c) is a local max/min.
C is incorrect because a point of inflection is when the second derivative is equal to zero and the function changes concavity at x = c
D is incorrect because the question states that f(x) is cts and differntiable everywhere
If function f(x) is cts and differentiable on the interval [a, b] and there is some defined value c in between a and b, please define what the following things mean.
f'(c) > 0
f''(c) < 0
At point c,
the function must be increasing at point c since the derivative is greater than 0
the function must be concave down at point c since the second derivative of f(x) is less than 0
What is a differential equation?
A differential equation is an equation that relates one or more unknown functions and their derivatives.
Note: Any answer sach deems close enough to the correct answer receives full points
Which test would be the best to determine if the following series converges or diverges? Explain why one answer you did not choose as the correct answer is not applicable to this series
sigma n = 1 to infinity of
(2n)/(n!)
a) Test for divergence
b) Integral test
c) Ratio Test
d) None of the above
C --> Ratio Test!!
A is incorrect because L'hopitals rule is required and you can not take the derivative of a factorial function
B is incorrect because there is no possible way to integrate a factorial.
D is incorrect because there is a correct answer of C
Please explain what the
1) Mean Value Theorem of Integrals (avg value)
2) Mean Value Theorem for Derivatives
3) Extreme Value Theorem
4) Intermediate Value Theorem
SPECIAL NOTE: 200 points earned if 2/4 or 3/4 of the information above is defined correctly, 400 points are earned if all four are defined correctly
Any definitions which are reasonable and close to the literal meaning. (Subject to judge's discretion)
A particle is moving in a straight line according to the following equations
x(t) = 2t2 - t5
Is the particle slowing down or speeding up at time t = 3. Explain
The particle is speeding up because the velocity and the acceleration of the particle at t = 2 have the same signs.
v(t) = 4t - 5t4 --> v(3) <0
a(t) = 4 - 20t3 --> a(3) < 0
Please determine on what interval(s) is f(x) both decreasing and concave down.
f(x) = 2et + t2
There is no interval in which f(x) is both concave down or decreasing because f''(x) is always greater than 0 for any real value x and this f(x) is always concave up.
f''(x) = 2et + 2 which is always greater than 0 because et is never negative
Please determine f(2) for which f(x) > 0 for an real value x and f(0) = 1
dy/dx = x2/y
NOTE: Answer must be provided in decimal form (so that it is easier to see if your answer was written on the answer slide)
Calculator Active
f(2) = 2.517 or 2.516
f(x) = sqrt(((2x^3 + 3)/3))
C (constant of integration) = 3
Reminder: f(x) & Constant of Integration are not required to attain full points, only f(2) = 2.516/2.517 is needed
What is the area inside the circle r=4sinθ and outside the circle r=3?
which of the following would be the best technique to find the f'(x)?
f(x) = (x2)/(x5+x)
a) power rule
b) product rule
c) quotient rule
d) u-substitution
e) anti-power rule
C --> Quotient Rule
A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=2t6 − 9t5 − 60t + 4
What is the total distance traveled by the particle over the time interval 0≤t≤7
Calculator Active
86,291.222 units
v(t) = 12t5 - 45t4 - 60
the integral of the absolute value of v(t) dt from 0 to 7 is 86,291.222
if f'(x) = 7x4 - 3x2
at what values of x does f(x) have a local maximum and minimum. (please answer in decimal form)
Justify your answer
Calculator Active
Since f'(x) goes from positive to negative at x = -0.655 and thus
f(x) has a local maximum at x = -0.655
Since f'(x) goes from negative to positive at x = 0.655 and thus
f(x) has a local maximum at x = -0.655
Ana opened a bank account with $1000 that earns interest, compounded continuously, at an annual rate of r=0.02. Money can be withdrawn from the account at regular intervals, and no additions to the account can be made. The function P models the balance of the account, in dollars, at time t. Assume that Ana withdraws money from the account continuously at a rate of N dollars per year.
Which of the following differential equations best describes the relationship between P and t?
a) dP/dt = 0.02P
b) dp/dt = 0.02P + 1000
c) dp/dt = 0.02(P-N)
d) dp/dt = 0.02P - N
The correct answer is d) --> dp/dt = 0.02P - N
The derivative dp/dt gives the account balance rate of change. If there are no additions or withdrawals, the rate of change of the balance is dPdt=0.02P. Continuously withdrawing N dollars per year gives dPdt=0.02P−N.
Find the particular solution for the following diff eq.
dy/dx = x2ln(x)
f(1) = 0
y=(1/3)x^3 * ln(x) - (1/9)(x^3) + (10/9)
NOTE: 100 points are earned if the entire answer is correct except there is no C present or the C value is incorrect. (C = 10/9)
(College board grants 2/5 (40%) of the points if you forget the "C" thus, 40% of 300 is 120 and thus I can only award 100 points)
Water is flowing into a tank at a rate of W(t) = 5sin(cos(7t)) + tt measured in gallons/second.
Please determine W'(3) and interpret the meaning in context. Indicate units of measure.
Calculator Active
W'(3) is equal to 31.663 (or 31.664)
W'(3) is measured in gallons per second per second
or gallons per second^2
The rate at which the water is flowing is increasing at a rate of 31.663 gallons per second per second
If f(x) is a cts and differentiable function and
f'(c) = 0
f''(c) = 3
f'(c-1) = 1
f'(c+1) = -1
Explain why these set of condititions are impossible for the function f(x)
f'(c) = 0 & f''(c) = 3 > 0 proves that f(x) has a local minimum at x = c
f'(c) = 0, f'(c-1) = 1 and f'(c+1) = -1 proves that f(x) has a local maximum at x = c.
Both can not be true at the same time, and thus these set of conditions are impossible for any function
IF dy/dx = x2y and f(0) = e What is....
1) limit as x approaches negative infinity of f(x)
2) The solution to the differential equation
NOTE #2: If answer #1 is incorrect, a total of 0 points are earned for the team.
1) the limit as x approaches negative infinity is ZERO
2) the limit as x approaches the negative infinity of f(x) is equal to e^-infinity which is 0.
NOTE: 200 points are earned if the entire answer (1 AND 2) is correct except there is no C present, or the C value is incorrect. (C = 1)
(College board grants 2/5 (40%) of the points if you forget the "C" thus, 40% of 400 is 160 and thus I can only award 200 points)
Write the 3rd degree Maclaurin polynomial for tan(x)
P(x) = x + (1/3)x3 + (2/15)x5
D --> (1.7,2)
Solution: https://ibb.co/rcbHcJB
Determine the following
limit as x approaches 0 of
3x2/sin(x2)
Note: Double points are earned for determining all critical points for the function f(x) = (3x^2)/(sin(x^2)
The use of the L'hopitals rule will get you to an answer of 3
A critical point is when f'(x) = 0 or DNE, when x = 0, f'(x) does not exist and is in indeterminate form of "0/0" Thus, f(x) has only one critical point at x = 0
The position, in centimeters, of a projectile, is modeled by x(t)=(1/10)t5 - (5/4)t4 + 6t3 where t is measured in seconds and x(0) = 0
What is the projectile’s maximum acceleration on the time interval 0≤t≤4? At what time t is this maximum acceleration attained?
Indicate units of measure
The maximum acceleration of the projectile is 32 centimeters per second per second and occurs at t=4 seconds.
Solution:
1) Differentiate x(t) twice to get
a(t) = 2t3−15t2+36t
D --> I & II
The only way to get this is to guess and check
Solution Explanation: https://ibb.co/n8WCybw
What is the average value of the function f(x) = sin(x^2) on the interval [0,pi/2] with the first three terms
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NOTE: Hardest Question on the Board -- by far
If a calc ab student gets the answer right... 3000 points
If a calc bc student gets it right 1500 points
The answer is 0.534
Solution: https://ibb.co/yVKrF5r