BSc CSIT 1st Semester Mathematics Solution PDF | TU Complete Guide (MTH117)
Introduction
Mathematics (MTH117) is the subject that separates students who struggle through the first semester from those who build real technical confidence. It's not the hardest subject in the BSc CSIT program in absolute terms — but it's the one that gives most students the most trouble early, for a simple reason: university calculus moves faster and goes deeper than anything in +2, and the first few weeks feel overwhelming before the pattern clicks.
The single most effective thing you can do for MTH117 — more effective than reading theory repeatedly — is working through solved problems. Seeing how a limit is evaluated, how an ODE is set up and solved, how a partial derivative is applied to find a minimum — those processes become clear through worked examples in ways that definitions and theorems alone don't produce.
This page is a complete MTH117 resource hub for TU BSc CSIT 1st Semester students. It includes the official course overview, chapter-wise topic summaries with original worked examples, the formula sheet, the most important TU exam questions, study tips, and the PDF download link. Whether you use the PDF, the examples on this page, or both — the goal is the same: understanding the method, not just reading about it.
Official Course Information
| Particular | Details |
|---|---|
| Course Title | Mathematics I |
| Course Code | MTH117 |
| Semester | First Semester |
| Nature | Theory |
| Full Marks | 80 + 20 |
| Pass Marks | 32 + 8 |
| Credit Hours | 3 |
| Primary Text | Stewart's Calculus: Early Transcendentals, 7th Edition |
Why Students Struggle with MTH117 (And How to Fix It)
Three patterns account for most MTH117 exam failures:
1. Reading instead of solving. Calculus is a doing-subject, not a reading-subject. A student who has read the integration chapter three times but worked zero practice problems will fail the integration question. A student who has worked fifteen integration problems may not remember reading anything, but will solve it correctly.
2. Memorizing formulas instead of understanding derivations. MTH117 exams frequently ask "derive" questions. A student who only memorized v = u + at (wrong course, but the point holds) without understanding why it's true cannot derive anything. Knowing where a formula comes from makes it impossible to forget and makes "derive" questions free marks.
3. Skipping the hard units. Differential equations (Unit 7) and partial derivatives (Unit 10) are exactly what most students deprioritize because they're confusing. They're also exactly what appears in long-answer sections because they carry the most teaching hours. Avoiding them costs the most marks.
The solution book/resource helps with pattern 1 by showing you worked processes. The formula sheet helps with patterns 1 and 2. The exam question list and study tips address pattern 3 directly.
What the Mathematics Solution Resource Covers
The MTH117 solution resource covers all 10 official units:
✔ Unit 1: Functions of One Variable ✔ Unit 2: Limits and Continuity ✔ Unit 3: Derivatives ✔ Unit 4: Applications of Derivatives ✔ Unit 5: Antiderivatives (Integration) ✔ Unit 6: Applications of Antiderivatives ✔ Unit 7: Ordinary Differential Equations ✔ Unit 8: Infinite Sequences and Series ✔ Unit 9: Plane and Space Vectors ✔ Unit 10: Partial Derivatives and Multiple Integration
For each unit: concept summary, key formulas, and step-by-step solved examples.
Chapter-Wise Topic Summaries and Solved Examples
Unit 1 & 2: Functions, Limits, and Continuity
Functions are the basic object that calculus operates on. Limits describe the behavior of a function as its input approaches a value — not what the function equals there, but what it approaches.
Most important limit to know: lim(x→0) [sin x / x] = 1
This appears directly in exam short questions and is used to evaluate other limits.
Worked Example — Evaluate lim(x→2) (x² − 4)/(x − 2)
Direct substitution gives 0/0 — indeterminate form. Factor the numerator:
(x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = (x + 2)
Now take the limit: lim(x→2) (x + 2) = 4
Worked Example — L'Hôpital's Rule: Evaluate lim(x→0) (e^x − 1)/x
This gives 0/0 form. Apply L'Hôpital (differentiate numerator and denominator separately):
lim(x→0) e^x / 1 = e^0 = 1
Asymptotes for rational functions:
- Vertical: where denominator = 0 (and numerator ≠ 0)
- Horizontal: compare degrees of numerator and denominator
Unit 3 & 4: Derivatives and Applications
The derivative of f(x) at a point measures the instantaneous rate of change — the slope of the tangent line.
Core differentiation rules:
- Power: d/dx(xⁿ) = nxⁿ⁻¹
- Product: (uv)' = u'v + uv'
- Quotient: (u/v)' = (u'v − uv')/v²
- Chain: d/dx[f(g(x))] = f'(g(x)) × g'(x)
Worked Example — Find dy/dx if y = x² sin(x)
Using product rule (u = x², v = sin x):
dy/dx = 2x · sin(x) + x² · cos(x) = x(2 sin x + x cos x)
Worked Example — Newton's Method (3 iterations to find root of f(x) = x³ − 2)
Formula: x_{n+1} = x_n − f(x_n)/f'(x_n)
f(x) = x³ − 2, f'(x) = 3x²
Start: x₀ = 1.5
| n | x_n | f(x_n) | f'(x_n) | x_{n+1} |
|---|---|---|---|---|
| 0 | 1.5 | 1.375 | 6.75 | 1.2963 |
| 1 | 1.2963 | 0.1782 | 5.040 | 1.2609 |
| 2 | 1.2609 | 0.0057 | 4.771 | 1.2597 |
Answer approaches ∛2 ≈ 1.2599 ✓
Optimization — standard setup:
- Express the quantity to maximize/minimize as a function of one variable
- Differentiate and set equal to zero: f'(x) = 0
- Verify using second derivative test (f'' > 0 → minimum, f'' < 0 → maximum)
Unit 5 & 6: Integration and Applications
Integration is the reverse of differentiation, and the Fundamental Theorem of Calculus is the bridge between them:
∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x)
Worked Example — Integration by Parts: Evaluate ∫ x·eˣ dx
Formula: ∫ u dv = uv − ∫ v du
Let u = x, dv = eˣ dx → du = dx, v = eˣ
∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C = eˣ(x − 1) + C
Worked Example — Simpson's Rule
Approximate ∫₁³ (1/x) dx using n = 4 subintervals:
h = (3−1)/4 = 0.5 x values: 1, 1.5, 2, 2.5, 3 f values: 1, 0.667, 0.5, 0.4, 0.333
Simpson's: (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + f₄] = (0.5/3)[1 + 4(0.667) + 2(0.5) + 4(0.4) + 0.333] = (0.1667)[1 + 2.667 + 1 + 1.6 + 0.333] = (0.1667)[6.6] = 1.0999
Exact value: ln(3) ≈ 1.0986 — excellent approximation.
Unit 7: Ordinary Differential Equations
ODEs are the most heavily weighted unit (6 hours) and the most commonly examined in long-answer format. Three methods you must know:
Method 1: Separable ODE Separate all y terms to one side, x terms to the other, then integrate both sides.
Worked Example: Solve dy/dx = xy
Separate: dy/y = x dx
Integrate: ln|y| = x²/2 + C
Solve: y = Ae^(x²/2) where A = e^C
Method 2: First-order Linear ODE (Integrating Factor)
Standard form: dy/dx + P(x)y = Q(x)
Integrating factor: μ = e^∫P(x)dx
Multiply through: d/dx[μy] = μQ(x)
Integrate both sides.
Worked Example: Solve dy/dx + 2y = 4
P(x) = 2, Q(x) = 4
μ = e^∫2dx = e^(2x)
Multiply: d/dx[e^(2x) y] = 4e^(2x)
Integrate: e^(2x) y = 2e^(2x) + C
y = 2 + Ce^(−2x)
Method 3: Second-order Homogeneous ODE
Form: ay'' + by' + cy = 0
Characteristic equation: ar² + br + c = 0
| Roots | Solution Form |
|---|---|
| Two distinct real roots r₁, r₂ | y = C₁e^(r₁x) + C₂e^(r₂x) |
| Repeated real root r | y = (C₁ + C₂x)e^(rx) |
| Complex roots α ± βi | y = e^(αx)[C₁cos(βx) + C₂sin(βx)] |
Worked Example: Solve y'' − 5y' + 6y = 0
Characteristic equation: r² − 5r + 6 = 0
Factor: (r − 2)(r − 3) = 0 → r = 2, r = 3
y = C₁e^(2x) + C₂e^(3x)
Unit 8: Infinite Sequences and Series
Key tests for convergence and the essential Maclaurin series:
Ratio Test: L = lim|a_{n+1}/a_n|. If L < 1: converges. L > 1: diverges. L = 1: inconclusive.
Essential Maclaurin Series (memorize these):
- e^x = 1 + x + x²/2! + x³/3! + ...
- sin x = x − x³/3! + x⁵/5! − ...
- cos x = 1 − x²/2! + x⁴/4! − ...
Worked Example: Test ∑ n/2ⁿ for convergence
a_n = n/2ⁿ, a_{n+1} = (n+1)/2^(n+1)
L = lim|(n+1)/2^(n+1) × 2ⁿ/n| = lim|(n+1)/2n| = 1/2 < 1
Series converges by ratio test.
Unit 9: Plane and Space Vectors
Essential formulas:
- Dot product: a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃
- Cross product magnitude: |a × b| = |a||b|sinθ
- a · b = 0 → vectors are perpendicular
- |a × b| = 0 → vectors are parallel
Worked Example: Find angle between a = [1,2,3] and b = [2,−1,1]
a · b = (1)(2) + (2)(−1) + (3)(1) = 2 − 2 + 3 = 3
|a| = √(1+4+9) = √14, |b| = √(4+1+1) = √6
cosθ = 3/(√14 × √6) = 3/√84 = 3/(2√21)
θ = arccos(3/2√21) ≈ 70.9°
Unit 10: Partial Derivatives and Multiple Integration
Partial derivatives treat all variables as constant except the one being differentiated with respect to.
Worked Example: Find ∂f/∂x and ∂f/∂y for f(x,y) = x²y + 3xy² − y³
∂f/∂x = 2xy + 3y² (treat y as constant)
∂f/∂y = x² + 6xy − 3y² (treat x as constant)
Second derivative test for critical points: Compute D = f_{xx}·f_{yy} − (f_{xy})²
- D > 0 and f_{xx} > 0 → local minimum
- D > 0 and f_{xx} < 0 → local maximum
- D < 0 → saddle point
Double integral example: ∫₀¹ ∫₀² xy dx dy = ∫₀¹ [x²y/2]₀² dy = ∫₀¹ 2y dy = [y²]₀¹ = 1
Complete Formula Sheet
Differentiation
| Rule | Formula |
|---|---|
| Power | d/dx(xⁿ) = nxⁿ⁻¹ |
| Product | (uv)' = u'v + uv' |
| Quotient | (u/v)' = (u'v − uv')/v² |
| Chain | d/dx[f(g(x))] = f'(g(x))·g'(x) |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
Integration
| Integral | Result |
|---|---|
| ∫xⁿ dx | xⁿ⁺¹/(n+1) + C |
| ∫eˣ dx | eˣ + C |
| ∫sin x dx | −cos x + C |
| ∫cos x dx | sin x + C |
| ∫1/x dx | ln |
| ∫u dv | uv − ∫v du (parts) |
Key Theorems
| Theorem | Statement |
|---|---|
| Mean Value Theorem | f'(c) = [f(b)−f(a)]/(b−a) |
| Fundamental Theorem | ∫ₐᵇ f(x)dx = F(b)−F(a) |
| L'Hôpital's Rule | lim f/g = lim f'/g' (for 0/0 or ∞/∞) |
| Newton's Method | x_{n+1} = x_n − f(x_n)/f'(x_n) |
ODE Methods
| Type | Method | Key Formula |
|---|---|---|
| Separable | Separate variables, integrate | dy/g(y) = f(x)dx |
| Linear 1st order | Integrating factor | μ = e^∫P dx |
| 2nd order homogeneous | Characteristic equation | ar²+br+c = 0 |
Numerical Integration
| Method | Formula |
|---|---|
| Trapezoidal | (h/2)[f₀ + 2f₁ + 2f₂ + ... + fₙ] |
| Simpson's | (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + ... + fₙ] (n even) |
Most Important TU Exam Questions
Almost Certain Every Year
- Evaluate a limit involving 0/0 or ∞/∞ form using L'Hôpital's Rule
- Apply Newton's Method — show 2–3 iterations for a given f(x)
- Apply Simpson's Rule or Trapezoidal Rule to given data
- Solve a first-order ODE (separable or linear with integrating factor)
- Solve a second-order homogeneous ODE using characteristic equation
- Find dot product and angle between two given vectors
- Find area between two curves using definite integration
- Apply integration by parts to evaluate a given integral
Frequently Appears
- Verify Mean Value Theorem for a given function on [a, b]
- Find local maxima/minima using first and second derivative tests
- Write Taylor or Maclaurin series for eˣ, sin x, or cos x
- Test a given series for convergence using the ratio test
- Find partial derivatives ∂f/∂x and ∂f/∂y and classify critical points
- Evaluate a double integral over a rectangular region
- Find the equation of a tangent line at a given point on a curve
Exam Preparation Tips
Work examples before reading — not after. Try each worked example yourself first. Compare your approach to the solution. The gap between what you tried and what the solution does is exactly what you need to learn.
Simpson's Rule and Newton's Method are procedural. Learn each as a numbered sequence of steps, not as a formula. Both appear almost every year and both are straightforward if you've practiced the procedure.
ODE methods are the most marks-dense skill in the course. If you can reliably solve separable, first-order linear, and second-order homogeneous ODEs, you're covering 6 teaching hours' worth of material that appears as long questions.
Show all steps in the exam. TU markers award step marks. A correct Newton's Method setup with an arithmetic error earns more marks than a correct final answer with no working shown.
Prepare the formula sheet from memory. Write out the differentiation rules, integration formulas, and the ODE method summary once per day in the final week. Active recall cements formulas better than passive re-reading.
Frequently Asked Questions
Is BSc CSIT 1st Semester Mathematics (MTH117) difficult? It's challenging because it covers a wide range of topics quickly and moves faster than +2 mathematics. Students who practice problem-solving daily rather than only reading theory consistently find it more manageable than those who study passively.
Which chapters carry the most marks? ODEs (Unit 7, 6 hours), Antiderivatives (Unit 5, 5 hours), and Applications (Unit 6, 5 hours) are typically heaviest in the exam. Vectors (Unit 9) and Applications of Derivatives (Unit 4) also generate full long questions. See the MTH117 Important Questions & Past Paper Guide for a complete priority breakdown.
How many solved examples should I practice before the exam? Aim for at least 5–6 problems per unit for the high-priority units (ODEs, Integration, Derivatives). For lower-priority units, 2–3 problems to understand the pattern is usually sufficient.
Is Stewart's Calculus enough, or do I need other books? Stewart (7th Edition) is the official text and covers everything in the syllabus. For additional worked examples, Piskunov's Differential and Integral Calculus and Thomas' Calculus (12th Edition, the official reference) are recommended supplements.
What is the pass mark for MTH117? Pass marks are 32 out of 80 (theory) + 8 out of 20 (internal). Total minimum: 40 out of 100.
Is there a solution manual for Stewart's Calculus? The official Student Solutions Manual (SSM) accompanies Stewart's text. It contains worked solutions for odd-numbered exercises. For TU-specific exam preparation, past papers and question banks aligned with the MTH117 syllabus are more directly useful.
Which is more important for the exam — theory or numericals? Both appear, but numerical problem-solving typically carries more marks in MTH117. Long questions almost always require applying a technique (evaluating an integral, solving an ODE, applying Newton's Method). Short questions test definitions and short applications. Neither can be entirely neglected.
Download the MTH117 Mathematics Resource PDF
The PDF resource below includes:
✔ Chapter-wise topic summaries for all 10 units ✔ Step-by-step worked solutions aligned with TU exam patterns ✔ Formula quick-reference sheet ✔ Important TU examination questions by unit ✔ 30-day and 7-day study plans
Download BSc CSIT Mathematics MTH117 Solution Resource PDF (Link to your actual PDF upload — replace this placeholder with your real download URL)
Conclusion
MTH117 is a subject that rewards students who engage with it actively — working problems, attempting derivations from memory, and checking their steps against worked solutions. The formula sheet and worked examples on this page are a starting point, not a substitute for that process.
Use the PDF for practice problems. Use the Complete MTH117 Guide for concept explanations and derivations. Use the MTH117 Important Questions & Past Paper Analysis to prioritize your revision in the final week.
For other 1st semester subjects: C Programming (CSC115) Notes, Digital Logic (CSC116) Notes, Physics (PHY118) Notes, and Introduction to Information Technology (CSC114) Notes.
In this page you can find BSC CSIT 1st semester Mathematics book solution full free download for students.You can download pdf file below.
){kind=link}
0 Comments