Mathematics IA Worked Examples CALCULUS: SUMMATION, INTEGRATION … · Mathematics IA Worked...
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Mathematics IA Worked Examples CALCULUS: SUMMATION, INTEGRATION AND THE FUNDAMENTAL THEOREM OF CALCULUS Produced by the Maths Learning Centre, The University of Adelaide. May 3, 2013 The questions on this page have worked solutions and links to videos on the following pages. Click on the link with each question to go straight to the relevant page. Questions 1. See Page 3 for worked solutions. Calculate the following sums: (a) 3 X k=1 3 X m=k mk ! (b) 100 X i=51 (2i - 1) (c) 99 X j =4 p j - 101 X j =6 p j - 1 2. See Page 5 for worked solutions. Given that ∑ 5 i=1 a i = 7, ∑ 12 i=6 a i = 25 and ∑ 13 i=2 b i = -4, calculate the following sums: (a) 12 X i=1 (1 - a i ) (b) 12 X i=1 (a i +2b i+1 ). 3. See Page 7 for worked solutions. The cubic function f (x)=2x 3 - 6x 2 + 7 has turning points at x = 0 and x = 2. Find the upper and lower sums U 5 and L 5 for f on the interval [-1, 3]. 4. See Page 9 for worked solutions. Find Z 5 3 x(x - 2) dx by first calculating the upper and lower sums U n and L n . 1
Transcript of Mathematics IA Worked Examples CALCULUS: SUMMATION, INTEGRATION … · Mathematics IA Worked...
Mathematics IAWorked Examples
CALCULUS: SUMMATION,INTEGRATION AND THE
FUNDAMENTAL THEOREM OFCALCULUS
Produced by the Maths Learning Centre,The University of Adelaide.
May 3, 2013
The questions on this page have worked solutions and links to videos onthe following pages. Click on the link with each question to go straight tothe relevant page.
Questions
1. See Page 3 for worked solutions.Calculate the following sums:
(a)3∑
k=1
(3∑
m=k
mk
)(b)
100∑i=51
(2i− 1) (c)99∑j=4
√j −
101∑j=6
√j − 1
2. See Page 5 for worked solutions.Given that
∑5i=1 ai = 7,
∑12i=6 ai = 25 and
∑13i=2 bi = −4, calculate the
following sums:
(a)12∑i=1
(1− ai) (b)12∑i=1
(ai + 2bi+1) .
3. See Page 7 for worked solutions.The cubic function f(x) = 2x3− 6x2 + 7 has turning points at x = 0 andx = 2. Find the upper and lower sums U5 and L5 for f on the interval[−1, 3].
4. See Page 9 for worked solutions.
Find
∫ 5
3
x(x − 2) dx by first calculating the upper and lower sums Un
and Ln.
1
5. See Page 13 for worked solutions.
Suppose
∫ 7
−1f(x) dx = 16,
∫ 7
3
f(x) dx = 12 and
∫ 4
−1f(x) dx = 20.
Find
∫ 4
3
f(x) dx.
6. See Page 14 for worked solutions.An antiderivative of the function f(x) = cos x− cos3 x is F (x) = 1
3sin3 x.
Use this fact and the Fundamental Theorem of Calculus to find∫ π2
π6
cosx− cos3 x dx.
7. See Page 15 for worked solutions.Let f(x) = 3x2 − 10x − 1. At what point(s) in the interval [1, 3] is thevalue of f(x) equal to its average value on [1, 3]?
8. See Page 17 for worked solutions.Find the area enclosed between the graphs of y = 2x2 − 5x + 3 andy = 3 + x− x2.
9. See Page 19 for worked solutions.
Let F (x) =
∫ x
0
e5−t2
dt and G(x) =
∫ x
−1e5−t
2
dt.
(a) Which is greater: F (2) or G(2)?
(b) Which is greater: F ′(2) or G′(2)?
10. See Page 20 for worked solutions.
Let G(x) =
∫ 3x
x
√sin t dt with domain [0, π
3]. Find G′(π
6).
1. Click here to go to question list.Calculate the following sums:
(a)3∑
k=1
(3∑
m=k
mk
)(b)
100∑i=51
(2i− 1) (c)99∑j=4
√j −
101∑j=6
√j − 1
Link to video on YouTube
2. Click here to go to question list.Given that
∑5i=1 ai = 7,
∑12i=6 ai = 25 and
∑13i=2 bi = −4, calculate the
following sums:
(a)12∑i=1
(1− ai) (b)12∑i=1
(ai + 2bi+1) .
Link to video on YouTube
3. Click here to go to question list.The cubic function f(x) = 2x3− 6x2 + 7 has turning points at x = 0 andx = 2. Find the upper and lower sums U5 and L5 for f on the interval[−1, 3].
Link to video on YouTube
4. Click here to go to question list.
Find
∫ 5
3
x(x − 2) dx by first calculating the upper and lower sums Un
and Ln.
Link to video on YouTube
5. Click here to go to question list.
Suppose
∫ 7
−1f(x) dx = 16,
∫ 7
3
f(x) dx = 12 and
∫ 4
−1f(x) dx = 20.
Find
∫ 4
3
f(x) dx.
Link to video on YouTube
6. Click here to go to question list.An antiderivative of the function f(x) = cos x− cos3 x is F (x) = 1
3sin3 x.
Use this fact and the Fundamental Theorem of Calculus to find∫ π2
π6
cosx− cos3 x dx.
Link to video on YouTube
7. Click here to go to question list.Let f(x) = 3x2 − 10x − 1. At what point(s) in the interval [1, 3] is thevalue of f(x) equal to its average value on [1, 3]?
Link to video on YouTube
8. Click here to go to question list.Find the area enclosed between the graphs of y = 2x2 − 5x + 3 andy = 3 + x− x2.
Link to video on YouTube
9. Click here to go to question list.
Let F (x) =
∫ x
0
e5−t2
dt and G(x) =
∫ x
−1e5−t
2
dt.
(a) Which is greater: F (2) or G(2)?
(b) Which is greater: F ′(2) or G′(2)?
Link to video on YouTube
10. Click here to go to question list.
Let G(x) =
∫ 3x
x
√sin t dt with domain [0, π
3]. Find G′(π
6).
Link to video on YouTube
