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Mat1512 Assignment 3 Solutions 2026

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Mat1512 Assignment 3 Solutions 2026 0-7-9-3-2-2-6-4-2-7 UNISA University of South Africa College of Science, Engineering and Technology Department of Mathematical Sciences Tutorial Letter 103/3/2026 MAT1512 – Calculus I Assignment 03 Opened: Thursday, 1 January 2026, 8:00 AM Due: Monday, 27 July 2026, 8:00 PM

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Mat1512 Assignment 3 Solutions 2026
UNISA
University of South Africa College of Science,
Engineering and Technology Department of
Mathematical Sciences Tutorial Letter 103/3/2026
MAT1512 – Calculus I Assignment 03
Opened: Thursday, 1 January 2026, 8:00 AM
Due: Monday, 27 July 2026, 8:00 PM

, QUESTION 1

Differentiate the following.

(a) 𝒚 = 𝒙𝟑 𝐥𝐧⁡ 𝒙(4 Marks)

Identify the rule

Since there are two functions multiplied together,

𝑢 = 𝑥 3 , 𝑣 = ln⁡ 𝑥


Use the Product Rule
𝑑𝑦
= 𝑢′ 𝑣 + 𝑢𝑣 ′
𝑑𝑥


Differentiate each function

Differentiate 𝑥 3 :

𝑢′ = 3𝑥 2


Differentiate ln 𝑥:
1
𝑣′ =
𝑥
Substitute into the Product Rule
𝑑𝑦 1
= (3𝑥 2 )(ln⁡ 𝑥) + (𝑥 3 ) ( )
𝑑𝑥 𝑥


Step 4: Simplify

Since
1
𝑥3 ( ) = 𝑥2,
𝑥


then

𝑑𝑦
= 3𝑥 2 ln⁡ 𝑥 + 𝑥 2
𝑑𝑥


𝑑𝑦
= 𝑥 2 (3ln⁡ 𝑥 + 1)
𝑑𝑥

Table of contents

  1. 01 QUESTION 1: Differentiate the following 2
    1. (a) y = x³ ln x 2
    2. Identify the rule 2
    3. Use the Product Rule 2
    4. Differentiate each function 2
    5. Substitute into the Product Rule 2
    6. Simplify 2
    7. (b) y = e²ˣ sin x 3
    8. Use Product Rule 3
    9. Differentiate each function 3
    10. Substitute 3
    11. Factorise 3
    12. (c) y = ln((x² + 1)/x) 4
    13. Method 1 (Best Method): Use the logarithm law 4
    14. Differentiate each term 4
    15. Combine 5
    16. Simplify into one fraction 5
  2. 02 QUESTION 2: Implicit differentiation 6
    1. (a) Find dy/dx given x² + xy + y² = 7 6
    2. Differentiate both sides 6
    3. Collect the derivative terms 7
    4. Factor out dy/dx 7
    5. Divide both sides 7
    6. (b) Determine the slope at (1,2) 7
    7. Substitute x = 1, y = 2 7
    8. Calculate numerator 7
    9. Calculate denominator 7
  3. 03 QUESTION 3: Critical points and second derivative test 8
    1. (a) Determine the critical points for f(x) = x³ - 6x² + 9x 8
    2. Step 1: Differentiate 8
    3. Factorise 8
    4. Set derivative equal to zero 8
    5. Find the corresponding y-values 8
    6. (b) Second Derivative Test 9
    7. Differentiate again 9
    8. At x = 1 9
    9. At x = 3 9
  4. 04 QUESTION 4: Optimization problem 10
    1. (a) Express the area as a function of x 10
    2. Step 1: Draw and identify the dimensions 10
    3. Write the formula for area 10
    4. Expand 10
    5. (b) Determine the dimensions that maximise the area 11
    6. To maximise the area, differentiate the area function 11
    7. Differentiate 11
    8. Set the derivative equal to zero 11
    9. Solve for x 11
    10. Find the height 12
    11. Verify that it is a maximum 12
    12. Maximum area (optional calculation) 12
  5. 05 QUESTION 5: Mean Value Theorem verification 13
    1. State the Mean Value Theorem 13
    2. Verify the conditions 13
    3. Check continuity and differentiability 13
    4. Calculate the average rate of change 14
    5. The interval is a = 1, b = e 14
    6. Compute f(e) and f(1) 14
    7. Substitute into the MVT formula 14
    8. Find the derivative 14
    9. Differentiate f(x) = ln x 14
    10. Apply the Mean Value Theorem 14
    11. According to the theorem 14
    12. Substitute f'(c) = 1/c 15
    13. Solve for c 15
    14. Multiply both sides by c(e-1) 15
    15. Verify that c is in the interval 15
    16. Check that c = e - 1 is in (1, e) 15

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