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Quiz 2: Fundamental Theorem of Calculus and Riemann Sums

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Suppose that f is continuous everywhere and that
∫ 5 ∫ 5 f(x) dx = - 6, 3f(x) dx = 6. 1 2
Find
∫ 1 2 f(x) dx.

 

Your answer is correct
Well done. We have ∫ 21 f(x) dx = -∫ 12 f(x) dx and also ∫ 15 f(x) dx -∫ 25 f(x) dx = ∫ 12 f(x) dx.
Not correct. You may try again.
Remember that ∫ ab f(x) dx = -∫ ba f(x) dx and that ∫ ab f(x) dx = ∫ ac f(x) dx + ∫ cb f(x) dx.
 

Question 2

 
 
Suppose that f and g are continuous everywhere and that
∫ ∫ 5 2 2 (f(x)+ 2g(x)) dx = - 3, 5 (f(x)- g(x)) = 1.
Find
∫ 5 2 (2f(x)- 5g(x)) dx.

 

Your answer is correct
Not correct. You may try again.
The data gives us ∫ 25(f + 2g) = -3 and ∫ 25(f - g) = -1. Also, 2f - 5g = 3(f - g) - (f + 2g). Now try the calculation again!
 

Question 3

 
 
The function g is a continuous function and a < b. Tick each correct statement in the list of options.
a)
∣ ∣ ∣∣∫ b ∣∣ ∫ b ∣∣ g(x) dx∣∣ ≤ ∣g(x)∣ dx a a
  b)
∣ ∣ ∣∣∫ b ∣∣ ∫ b ∣∣ g(x) dx∣∣ ≥ g(x) dx a a
c)
∣∣∫ b ∣∣ ∫ b ∣∣ g(x) dx∣∣ ≤ g(x) dx ∣ a ∣ a
  d)
∫ b ∫ b ∣g(x)∣ dx ≤ g(x) dx a a
e)
∫ b ∫ b a ∣g(x)∣ dx ≥ a g(x) dx

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be false.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
Your answers are correct
  1. True.
  2. True.
  3. False.
  4. False.
  5. True.
 

Question 4

 
 
A particle accelerates from rest in a straight line over a 6 second period. The graph of its acceleration, f(t) at time t, from t = 0 seconds to t = 6 seconds, is shown below. At what time after the start does the particle reach maximum velocity?
PIC

 

Your answer is correct
Well done! Whenever the graph of acceleration is positive, the velocity is increasing. Since the acceleration is zero at t = 5, this marks the time of maximum velocity.
Not correct. You may try again.
Remember that acceleration is the derivative of velocity, and that an increasing function has positive derivative.
 

Question 5

 
 
This question uses the same data as the previous question. A particle accelerates from rest in a straight line over a 6 second period. The graph of its acceleration f(t) at time t, from t = 0 seconds to t = 6 seconds is shown below.
PIC
Tick all the correct statements.
a) The average acceleration of the particle over these 6 seconds is 3.5m∕s2.   b) The velocity at time t = 4 is 6m∕s.
c) The average acceleration of the particle over these 6 seconds is 1.4m∕s2.   d) The distance travelled between t = 0 and t = 2 is 2 metres.
e) The average acceleration of the particle over these 6 seconds is 1.08˙3m∕s2.   f) The distance travelled between t = 0 and t = 2 is 4 3 metres.

 

There is at least one mistake.
For example, choice (a) should be false.
How is the average value of a function g(x) over an interval [a,b] calculated? How is it related to the definite integral ∫ ab g(x) dx ?
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be false.
How is the average value of a function g(x) over an interval [a,b] calculated? How is it related to the definite integral ∫ ab g(x) dx ?
There is at least one mistake.
For example, choice (d) should be false.
First obtain an expression for the velocity of the particle as a function of time, over the first two seconds of travel.
There is at least one mistake.
For example, choice (e) should be true.
There is at least one mistake.
For example, choice (f) should be true.
Your answers are correct
  1. False. How is the average value of a function g(x) over an interval [a,b] calculated? How is it related to the definite integral ∫ ab g(x) dx ?
  2. True.
  3. False. How is the average value of a function g(x) over an interval [a,b] calculated? How is it related to the definite integral ∫ ab g(x) dx ?
  4. False. First obtain an expression for the velocity of the particle as a function of time, over the first two seconds of travel.
  5. True.
  6. True.
 

Question 6

 
 
Find the value of
∫ 32π 0 ∣cos x∣ dx.
(A graph helps.)
a) 1 2   b) 1
c) 2π   d) 3
e) None of the above.

 

Not correct. Choice (a) is false.
Where does the graph of the cosine function cross the horizontal axis? This will give you a way of sketching the graph of ∣cosx ∣ . You can also make use of symmetry.
Not correct. Choice (b) is false.
Where does the graph of the cosine function cross the horizontal axis? This will give you a way of sketching the graph of ∣cosx∣ . You can also make use of symmetry.
Not correct. Choice (c) is false.
Where does the graph of the cosine function cross the horizontal axis? This will give you a way of sketching the graph of ∣cosx∣. You can also make use of symmetry.
Your answer is correct.
The graph of ∣cos x∣ shows that the required answer is ∫ π∕2 3 0 cosx dx , which equals 3.
Not correct. Choice (e) is false.
 

Question 7

 
 
Find the average value of g(t) over [0,3], where
{ 0 if 0 ≤ t ≤ 1 g(t) = 1 if 1 < t ≤ 3
a) 3   b) 3 2
c) 2 3   d) 5 2
e) None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 8

 
 
Which of the following expressions may be evaluated using the Fundamental Theorem of Calculus Part II? Tick all those that can be. (You do not have to evaluate them.)
a)
∫ 5 x--3--dx 1 x2 - 4
  b)
∫ 5 x- 3 -2----dx 1 x + 1
c)
∫ 1 1- -1 x dx
  d)
∫ 1 x -e----dx -1 ex + 1
e)
∫ 1 ex + e-x -ex --1-dx - 1
  f) None of the above

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be false.
There is at least one mistake.
For example, choice (d) should be true.
There is at least one mistake.
For example, choice (e) should be false.
There is at least one mistake.
For example, choice (f) should be false.
Your answers are correct
  1. False.
  2. True.
  3. False.
  4. True.
  5. False.
  6. False.
 

Question 9

 
 
Consider the sum
-1 + 1-+ -1 + .........+ -1-. 23 33 43 113
Tick each option that equals this sum.
a) The Riemann Lower Sum for f(x) = 1- x3 on the interval [2,10] .   b) The Riemann Upper Sum for f(x) =-13 x on the interval [1,10] .
c) The Riemann Lower Sum for 1 f(x) = x3 on the interval [1,11] .   d) The Riemann Upper Sum for -1 f(x) = x3 on the interval [2,12] .
e) None of the above.

 

There is at least one mistake.
For example, choice (a) should be false.
Remember that the Lower Sums use the minimum function values on each sub-interval.
There is at least one mistake.
For example, choice (b) should be false.
Remember that the Upper Sums use the maximum function values on each sub-interval.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be true.
There is at least one mistake.
For example, choice (e) should be false.
Your answers are correct
  1. False. Remember that the Lower Sums use the minimum function values on each sub-interval.
  2. False. Remember that the Upper Sums use the maximum function values on each sub-interval.
  3. True.
  4. True.
  5. False.
 

Question 10

 
 
Part of the graph of the derivative function df dx = f′(x) is shown below. Tick each correct option.
PIC
a) The minimum value of f(x) on [-1,4] is 0.   b) The function f is an increasing function on [-1,1].
c) f(4) > f(1).   d) The maximum value of f(x) on [-1,4.5] occurs when x = 4.
e) The function f is concave down on [-1,4.5].

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be true.
There is at least one mistake.
For example, choice (e) should be false.
Your answers are correct
  1. False.
  2. True.
  3. True.
  4. True.
  5. False.