Calculus – I

1️⃣ Plotting of Functions

Plotting a function means drawing its graph on a coordinate system. A function $y = f(x)$y=f(x) represents a rule that assigns each input $x$x a unique output $y$y.

Common Types of Functions:

📌 Tools like Desmos, GeoGebra, and a scientific calculator help visualize graphs easily.

2️⃣ Intuitive Idea of Continuous and Differentiable Functions

Continuity

A function is continuous if it has no breaks, jumps, or holes in its graph.

A function $f(x)$f(x) is continuous at x=ax=ax=a if:$$\lim_{x\to a} f(x) = f(a)$$x→alim​f(x)=f(a)

Differentiability

A function is differentiable if its graph has no sharp corners or vertical tangents.
It must also be continuous, but a continuous function need not always be differentiable.

Example:

• $|x|$∣x∣ is continuous everywhere but not differentiable at x=0x=0x=0 due to a sharp corner.

3️⃣ Plotting of Curves

Curve sketching involves understanding:

• Domain & Range
• Increasing/decreasing intervals
• Maximum & minimum points
• Concavity and points of inflection
• Asymptotes

The first derivative helps determine slopes and turning points.
The second derivative helps analyze concavity.

4️⃣ Approximation: Taylor and Binomial Series (Statements Only)

Taylor Series

A powerful method to approximate functions using polynomials.$$f(x) = f(a) + f'(a)(x-a) + \frac{f”(a)}{2!}(x-a)^2 + \cdots$$f(x)=f(a)+f′(a)(x−a)+2!f′′(a)​(x−a)2+⋯

Useful when a function is difficult to compute directly.

Binomial Series

For any real number $n$n, the expansion of $(1+x)^n$(1+x)n:$$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots$$(1+x)n=1+nx+2!n(n−1)​x2+⋯

5️⃣ First-Order Differential Equations

A First-Order Differential Equation involves the first derivative of $y$y, like:$$\frac{dy}{dx} + P(x)y = Q(x)$$dxdy​+P(x)y=Q(x)

Integrating Factor (IF)

Used to solve linear first-order differential equations.

Formula:$$IF = e^{\int P(x)\,dx}$$IF=e∫P(x)dx

Solution:$$y \cdot IF = \int Q(x)\cdot IF \, dx + C$$y⋅IF=∫Q(x)⋅IFdx+C

6️⃣ Second-Order Differential Equations

A Second-Order Differential Equation involves the second derivative of $y$y:$$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$$adx2d2y​+bdxdy​+cy=0

✧ Homogeneous Linear Equation with Constant Coefficients

Characteristic equation:$$ar^2 + br + c = 0$$ar2+br+c=0

Three cases:

1. Real and distinct roots $$y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$$y=C1​er1​x+C2​er2​x
2. Real and equal roots $$y = (C_1 + C_2 x)e^{rx}$$y=(C1​+C2​x)erx
3. Complex roots $$y = e^{\alpha x}(C_1\cos \beta x + C_2\sin \beta x)$$y=eαx(C1​cosβx+C2​sinβx)

7️⃣ Wronskian and General Solution

Wronskian helps determine whether two solutions are linearly independent.

For functions $y_1, y_2$y1​,y2​:$$W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1′ & y_2′ \end{vmatrix} \neq 0 \Rightarrow \text{Linearly Independent solutions}$$W(y1​,y2​)=​y1​y1′​​y2​y2′​​​=0⇒Linearly Independent solutions

General solution = Complementary Function + Particular Integral

8️⃣ Existence and Uniqueness Theorem (Statement Only)

If $P(x)$P(x) and $Q(x)$Q(x) are continuous near $x=a$x=a, then the initial value problem$$\frac{dy}{dx} = f(x,y),\quad y(a)=b$$dxdy​=f(x,y),y(a)=b

has a unique solution in a neighborhood of $a$a.

9️⃣ Particular Integral (P.I.)

When the differential equation is non-homogeneous:$$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = F(x)$$adx2d2y​+bdxdy​+cy=F(x)

The particular integral finds a specific solution depending on $F(x)$F(x).
Final solution:$$y = \text{C.F.} + \text{P.I.}$$y=C.F.+P.I.

Conclusion

Calculus – I forms the foundation for all advanced mathematics and applied sciences. Understanding concepts like continuity, differentiability, and differential equations not only builds theoretical knowledge but also enhances problem-solving and analytical skills.