Differential Calculus Revision Notes BTER Polytechnic 1st Semester

 

Differential Calculus

Differential calculus is concerned with the concept of a derivative, which represents the rate at which a function changes as its input changes. It's a fundamental branch of calculus used in various fields of engineering to model rates of change and to find approximations.

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2.1 Definition of Function; Graphs of $e^x$, $\log x$, and $x^n$

Definition of a Function:

A function is a relation between two sets where each input (from the domain) is associated with exactly one output (from the codomain). Mathematically, a function $f$ from set $A$ to set $B$ is denoted as $f : A \to B$, where $f(x) = y$, and each value of $x$ has one corresponding value of $y$.

• Domain: The set of all possible input values.
• Range: The set of all possible output values.

A function can be represented in various forms, such as algebraic expressions, graphs, tables, or verbal descriptions.

Graphs of Common Functions:

1. Exponential Function $e^x$:

   • The graph of $y = e^x$ is an increasing curve that passes through the point $(0, 1)$.
   • Key Properties:
     • Domain: $(-\infty, \infty)$
     • Range: $(0, \infty)$
     • Asymptote: The x-axis is a horizontal asymptote for large negative values of $x$.

2. Logarithmic Function $\log_a x$:

   • The graph of $y = \log_a x$ (where $a > 0, a \neq 1$) is the inverse of the exponential function $y = a^x$.
   • Key Properties:
     • Domain: $(0, \infty)$
     • Range: $(-\infty, \infty)$
     • Asymptote: The y-axis is a vertical asymptote.

3. Power Function $y = x^n$:

   • The graph of $y = x^n$ behaves differently depending on whether $n$ is even or odd.

   • For even $n$: The graph is symmetric about the y-axis and passes through the origin.

   • For odd $n$: The graph is symmetric about the origin.

   • Key Properties:

     • Domain: $(-\infty, \infty)$ for all integer values of $n$
     • Range: Depends on the value of $n$.

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2.2 Concept of Limits

Concept of Limits:

A limit describes the behavior of a function as its input approaches a particular value. The limit of $f(x)$ as $x$ approaches $a$ is denoted as:

$$\lim_{x \to a} f(x) = L$$

This means that as $x$ gets arbitrarily close to $a$, the function $f(x)$ gets arbitrarily close to $L$.

• Right-hand Limit: $\lim_{x \to a^+} f(x)$
• Left-hand Limit: $\lim_{x \to a^-} f(x)$

For the limit to exist, both the right-hand and left-hand limits must be equal.

Standard Limits:

Here are some standard limits that are essential in differential calculus:

1. Limit of Power Function:

   $$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$$

   This is useful when differentiating polynomial functions.

2. Sine Limit:

   $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

   This is a fundamental limit used in the derivation of derivatives of trigonometric functions.

3. Exponential Limit:

   $$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$$

   This is used when finding the derivative of exponential functions.

4. Logarithmic Limit:

   $$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$$

   This limit is crucial when differentiating logarithmic functions.

5. Binomial Limit:

   $$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$$

   This limit is essential for understanding the exponential function and its derivative.

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2.3 Differentiation of Trigonometric Functions

Differentiation of trigonometric functions involves applying standard differentiation rules. The derivatives of the basic trigonometric functions are:

1. Derivative of Sine: $$\frac{d}{dx} \sin x = \cos x$$
2. Derivative of Cosine: $$\frac{d}{dx} \cos x = -\sin x$$
3. Derivative of Tangent: $$\frac{d}{dx} \tan x = \sec^2 x$$
4. Derivative of Cotangent: $$\frac{d}{dx} \cot x = -\csc^2 x$$
5. Derivative of Secant: $$\frac{d}{dx} \sec x = \sec x \tan x$$
6. Derivative of Cosecant: $$\frac{d}{dx} \csc x = -\csc x \cot x$$

Applications:

• Physics and Engineering: The derivative of trigonometric functions is used to describe motion (velocity, acceleration), oscillations, waves, and alternating currents.
• Optimization: These derivatives are used in optimization problems involving periodic functions.

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2.4 Differentiation of Inverse Trigonometric Functions

The inverse trigonometric functions are the inverse of the basic trigonometric functions. Their derivatives are derived using implicit differentiation and the chain rule. Here are the derivatives of the common inverse trigonometric functions:

1. Derivative of $\sin^{-1} x$ (Arcsine):

   $$\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}, \quad -1 \leq x \leq 1$$

2. Derivative of $\cos^{-1} x$ (Arccosine):

   $$\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}, \quad -1 \leq x \leq 1$$

3. Derivative of $\tan^{-1} x$ (Arctangent):

   $$\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$$

4. Derivative of $\cot^{-1} x$ (Arccotangent):

   $$\frac{d}{dx} \cot^{-1} x = -\frac{1}{1 + x^2}$$

5. Derivative of $\sec^{-1} x$ (Arcsecant):

   $$\frac{d}{dx} \sec^{-1} x = \frac{1}{|x| \sqrt{x^2 - 1}}$$

   for $|x| > 1$

6. Derivative of $\csc^{-1} x$ (Arccosecant):

   $$\frac{d}{dx} \csc^{-1} x = -\frac{1}{|x| \sqrt{x^2 - 1}}$$

   for $|x| > 1$

Applications:

• Engineering: Inverse trigonometric functions are used in solving problems involving angles of elevation, depression, and trigonometric identities.
• Physics: They are used to find angles in problems related to oscillations, waves, and rotational motion.

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Summary of Key Formulas:

• Limits:

  • $\lim_{x \to 0} \frac{\sin x}{x} = 1$
  • $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$
  • $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$

• Differentiation of Trigonometric Functions:

  • $\frac{d}{dx} \sin x = \cos x$
  • $\frac{d}{dx} \cos x = -\sin x$
  • $\frac{d}{dx} \tan x = \sec^2 x$

• Differentiation of Inverse Trigonometric Functions:

  • $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$
  • $\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$

These concepts and formulas form the core foundation for solving problems involving differentiation in engineering mathematics.