FINAL EXAM MATH1050 (Functions-Differentiation-integration-sequences-matrices) chapters

Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

Part A

Each question in Part A is worth 2 marks. Write your answers in the space provided. Show full working.

1. Determine the domain and range of f(x) = √ x+ 2.

2. Determine the following limit, or show that it does not exist.

lim x→5

( x2 − 2x− 15

x− 5

) .

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

3. Determine the following limit, or show that it does not exist.

lim x→3

(√ x− 3 x+ 3

) .

4. Determine dy

dx for y = x2ex.

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

5. Determine

∫ (3x2 + 2)(x3 + 2x)2 dx.

6. Determine

∫ 2

x+ 7 dx.

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

7. The first three terms of an arithmetic sequence are −1, 2 and 5.

(a) Determine a closed form expression for the nth term. Simplify your answer.

(b) Determine the sum of the first 50 terms of the sequence.

8. Determine the 12th term of the geometric sequence 0.3, 0.9, 2.7, ...

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

9. Let A =

( −1 −1 −1 3

) (a) Determine A−1.

(b) Hence determine X if

( −1 −1 −1 3

) X =

( 1 3 2 5

) .

10. Let M =

( 2 −1 3 4

) and N =

( −3 2 6 −5

) . Determine MN .

End of Part A

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

Part B

There are 46 marks in Part B. Each of the following questions carries the stated number of marks. Write your answers in the space provided. Show full working. Part marks may be awarded for correct working.

1. Sketch a possible graph of the function f : R→ R with the following given properties: f(−2) = −1, f(1) = −3, f ′(−2) = f ′(1) = 0, f ′(x) < 0 whenever −2 < x < 1, f ′(x) > 0 whenever x < −2 or x > 1, lim

x→−∞ f(x) = −4,

lim x→4

f(x) =∞. (5 marks)

y

x

(4 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

2. Determine from first principles the derivative of f(x) = 2

x− 3 (for x 6= 3). (4 marks)

Note: f ′(x) = lim h→0

f(x+ h)− f(x) h

.

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

3. A person on a surf ski is 3 km out to sea from the nearest point of a straight beach. The person’s destination is 6 km along the beach . The fastest the person can paddle is 4 km/h and their maximum running speed is 5 km/h. How far along the beach should the person go ashore to reach their destination in the least possible time? (6 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

4. (a) Determine

∫ 2x− 1√

x dx. (2 marks)

(b) Determine

∫ 1 0

(ex + e−x)2 dx. (3 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

(c) Determine

∫ sin3 x

cos5 x dx. (Note:

d

dx (tanx) = sec2 x) (5 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

5. Consider the curve with equation x2 − 2xy2 + y3 = 2. Determine dy dx

. (4 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

6. Use Mathematical Induction to show that n∑

i=1

1

(3i− 1)(3i+ 2) =

n

6n+ 4 , for all integers n ≥ 1.

(6 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

7. The second term of a convergent geometric series is 8

5 . If the sum of the series is 10, determine

all possible first terms and common ratios. (6 marks)

continued...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

8. A group of ten people go to Lang Park to watch the Wallabies play Ireland, and each buys one drink during the game. Admission costs $49 for people 16 years and over and $25 for people aged under 16. The people who are 18 years old or over each buy a beer, those who are aged 16 or 17 buy a juice and those who are under 16 buy water. The drink costs are $6 for beer, $4 for juice, and $5 for water. Together they pay $370 for admission and $53 for drinks. Use matrices to determine how many people buy each type of drink.

Note: you can do this question without needing to find the inverse of a 3 × 3 matrix. (5 marks)

End of exam

Formulae Sheet...

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Semester One Practice Final Examination, 2020 MATH1050 Mathematical Foundations

Formulae Sheet

Cosine Rule: a2 = b2 + c2 − 2bc cosA

Sine Rule: sinA

a =

sinB

b =

sinC

c

sin(θ + φ) = sin θ cosφ+ cos θ sinφ

sin(θ − φ) = sin θ cosφ− cos θ sinφ

cos(θ + φ) = cos θ cosφ− sin θ sinφ

cos(θ − φ) = cos θ cosφ+ sin θ sinφ

cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1− 2 sin2 θ

sin 2θ = 2 sin θ cos θ

For an arithmetic sequence of the form {aj}nj=1, where aj = a+ (j − 1)d,

Sn = n∑

i=1

ai = n

2 (2a+ (n− 1)d).

For a geometric sequence of the form {aj}nj=1, where aj = arj−1,

Sn = n∑

i=1

ai = a(1− rn)

1− r = a(rn − 1) r − 1

.

For a convergent geometric series, S∞ = a

1− r , −1 < r < 1.

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