Continuity And Differentiability - Continuity At A Point

Continuity and Differentiability - Continuity at a point

Definition of Continuity at a point
Symbolic representation of continuity at a point: $$f(a)$$
If a function is continuous at a point, then $$\lim_{{x \to a}} f(x)$$ exists
Graphical representation of continuity at a point
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Continuity in an interval

Definition of Continuity in an interval
Symbolic representation of continuity in an interval: $$f(x)$$
If a function is continuous in an interval, then it is continuous at every point in that interval
Graphical representation of continuity in an interval
Example: $$f(x) = \sin(x)$$ in the interval $$[-\pi, \pi]$$

Continuity and Differentiability - Discontinuity

Definition of Discontinuity
Types of Discontinuity: Removable Discontinuity, Jump Discontinuity, and Infinite Discontinuity
Symbolic representation of discontinuity
Graphical representation of discontinuity
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Types of Discontinuity

Removable Discontinuity: when there is a hole in the graph of a function at a certain point but can be filled
Jump Discontinuity: when there exist two one-sided limits at a certain point but they are not equal
Infinite Discontinuity: when the function approaches infinity or negative infinity at a certain point
Graphical representation of each type of discontinuity
Example: $$f(x) = \frac{{\sin(x)}}{{x}}$$ at $$x = 0$$

Continuity and Differentiability - Differentiability

Definition of Differentiability
Symbolic representation of differentiability: $$f’(x)$$
Differentiability implies continuity
Differentiability at a point vs Differentiability in an interval
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Differentiation Rules

Power Rule: $$\frac{{d}}{{dx}}(x^n) = nx^{n-1}$$
Constant Multiple Rule: $$\frac{{d}}{{dx}}(cf(x)) = c \cdot \frac{{d}}{{dx}}(f(x))$$
Sum and Difference Rule: $$\frac{{d}}{{dx}}(f(x) \pm g(x)) = \frac{{d}}{{dx}}(f(x)) \pm \frac{{d}}{{dx}}(g(x))$$
Product Rule: $$\frac{{d}}{{dx}}(f(x) \cdot g(x)) = f’(x)g(x) + f(x)g’(x)$$
Quotient Rule: $$\frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{f’(x)g(x) - f(x)g’(x)}}{{g^2(x)}}$$

Continuity and Differentiability - Examples of Differentiation

Example: Differentiate $$y = x^3 - 2x^2 + 5x + 1$$
Example: Differentiate $$y = \sqrt{x} + \frac{1}{x}$$
Example: Differentiate $$y = \sin(x) + \cos(x)$$
Example: Differentiate $$y = e^x \ln(x)$$
Example: Differentiate $$y = \frac{{x^3 - 4}}{{x - 2}}$$

Continuity and Differentiability - Rolle’s Theorem

Statement of Rolle’s Theorem
Hypotheses of Rolle’s Theorem: Continuity and Differentiability in closed interval, and equality of function values at the endpoints
Conclusion of Rolle’s Theorem: There exists at least one point in the interval where the derivative is zero
Geometrical interpretation of Rolle’s Theorem
Example: Verify that the function $$f(x) = 2x^3 - 9x^2 + 12x + 7$$ satisfies Rolle’s Theorem in the interval $$[1, 3]$$

Continuity and Differentiability - Continuity at a point

Definition of Continuity at a point
Symbolic representation of continuity at a point: $$f(a)$$
If a function is continuous at a point, then $$\lim_{{x \to a}} f(x)$$ exists
Graphical representation of continuity at a point
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$
Substitute $$x = 2$$ in the function: $$f(2) = \frac{{2^2 - 4}}{{2 - 2}} = \frac{0}{0}$$
Simplify the equation: $$f(2) = \frac{0}{0}$$ does not produce a valid solution. Therefore, the function is not continuous at $$x = 2$$
Explain the concept with a graphical representation of the function and the vertical asymptote at $$x = 2$$

Continuity and Differentiability - Continuity in an interval

Definition of Continuity in an interval
Symbolic representation of continuity in an interval: $$f(x)$$
If a function is continuous in an interval, then it is continuous at every point in that interval
Graphical representation of continuity in an interval
Example: $$f(x) = \sin(x)$$ in the interval $$[-\pi, \pi]$$
Explain that the function is continuous at every point in the interval
Show the graphical representation of the sine function and explain that it is smooth and continuous in the given interval

Continuity and Differentiability - Discontinuity

Definition of Discontinuity
Types of Discontinuity: Removable Discontinuity, Jump Discontinuity, and Infinite Discontinuity
Symbolic representation of discontinuity
Graphical representation of discontinuity
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$
Substitute $$x = 2$$ in the function: $$f(2) = \frac{{2^2 - 4}}{{2 - 2}} = \frac{0}{0}$$
Simplify the equation: $$f(2) = \frac{0}{0}$$ does not produce a valid solution. Therefore, the function has a removable discontinuity at $$x = 2$$
Explain the concept with a graphical representation of the function and the hole in the graph at $$x = 2$$

Continuity and Differentiability - Types of Discontinuity

Removable Discontinuity: when there is a hole in the graph of a function at a certain point but can be filled
Jump Discontinuity: when there exist two one-sided limits at a certain point but they are not equal
Infinite Discontinuity: when the function approaches infinity or negative infinity at a certain point
Graphical representation of each type of discontinuity
Example: $$f(x) = \frac{{\sin(x)}}{{x}}$$ at $$x = 0$$
Substitute $$x = 0$$ in the function: $$f(0) = \frac{{\sin(0)}}{{0}} = \frac{0}{0}$$
Simplify the equation: $$f(0) = \frac{0}{0}$$ does not produce a valid solution. Therefore, the function has an infinite discontinuity at $$x = 0$$
Explain the concept with a graphical representation of the function and the vertical asymptote at $$x = 0$$

Continuity and Differentiability - Differentiability

Definition of Differentiability
Symbolic representation of differentiability: $$f’(x)$$
Differentiability implies continuity
Differentiability at a point vs Differentiability in an interval
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$
Differentiate the function with respect to $$x$$ using the quotient rule: $$f’(x) = \frac{{(x - 2)(2x) - (x^2 - 4)(1)}}{{(x - 2)^2}}$$
Simplify the expression: $$f’(x) = \frac{{2x^2 - 4x - 2x^2 + 4}}{{(x - 2)^2}}$$
Further simplify the expression: $$f’(x) = \frac{{8 - 4x}}{{(x - 2)^2}}$$
Substitute $$x = 2$$ in the derivative: $$f’(2) = \frac{{8 - 4(2)}}{{(2 - 2)^2}} = \frac{0}{0}$$
Simplify the equation: $$f’(2) = \frac{0}{0}$$ does not produce a valid solution. Therefore, the function is not differentiable at $$x = 2$$

Continuity and Differentiability - Differentiation Rules

Power Rule: $$\frac{{d}}{{dx}}(x^n) = nx^{n-1}$$
Constant Multiple Rule: $$\frac{{d}}{{dx}}(cf(x)) = c \cdot \frac{{d}}{{dx}}(f(x))$$
Sum and Difference Rule: $$\frac{{d}}{{dx}}(f(x) \pm g(x)) = \frac{{d}}{{dx}}(f(x)) \pm \frac{{d}}{{dx}}(g(x))$$
Product Rule: $$\frac{{d}}{{dx}}(f(x) \cdot g(x)) = f’(x)g(x) + f(x)g’(x)$$
Quotient Rule: $$\frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{f’(x)g(x) - f(x)g’(x)}}{{g^2(x)}}$$
Example: Differentiate $$y = x^3 - 2x^2 + 5x + 1$$
Example: Differentiate $$y = \sqrt{x} + \frac{1}{x}$$
Example: Differentiate $$y = \sin(x) + \cos(x)$$
Example: Differentiate $$y = e^x \ln(x)$$
Example: Differentiate $$y = \frac{{x^3 - 4}}{{x - 2}}$$

Continuity and Differentiability - Examples of Differentiation

Example: Differentiate $$y = x^3 - 2x^2 + 5x + 1$$
- Apply the power rule: $$\frac{{dy}}{{dx}} = 3x^2 - 4x + 5$$
Example: Differentiate $$y = \sqrt{x} + \frac{1}{x}$$
- Apply the sum rule and the quotient rule: $$\frac{{dy}}{{dx}} = \frac{1}{2\sqrt{x}} - \frac{1}{x^2}$$
Example: Differentiate $$y = \sin(x) + \cos(x)$$
- Apply the sum rule and the derivative of sine and cosine: $$\frac{{dy}}{{dx}} = \cos(x) - \sin(x)$$
Example: Differentiate $$y = e^x \ln(x)$$
- Apply the product rule and the derivative of exponential and logarithmic functions: $$\frac{{dy}}{{dx}} = e^x \ln(x) + \frac{e^x}{x}$$
Example: Differentiate $$y = \frac{{x^3 - 4}}{{x - 2}}$$
- Simplify the function and apply the quotient rule: $$\frac{{dy}}{{dx}} = \frac{{2x^3 - 8}}{{(x - 2)^2}}$$

Continuity and Differentiability - Rolle’s Theorem

Statement of Rolle’s Theorem
Hypotheses of Rolle’s Theorem: Continuity and Differentiability in closed interval, and equality of function values at the endpoints
Conclusion of Rolle’s Theorem: There exists at least one point in the interval where the derivative is zero
Geometrical interpretation of Rolle’s Theorem
Example: Verify that the function $$f(x) = 2x^3 - 9x^2 + 12x + 7$$ satisfies Rolle’s Theorem in the interval $$[1, 3]$$
- Check the conditions of Rolle’s Theorem:
  - Continuity: The function is a polynomial, so it is continuous in the interval $$[1, 3]$$
  - Differentiability: The function is a polynomial, so it is differentiable in the interval $$[1, 3]$$
  - Equality of function values at the endpoints: $$f(1) = -2$$ and $$f(3) = 34$$
- Since all the conditions are satisfied, we can conclude that there exists at least one point in the interval $$[1, 3]$$ where the derivative is zero

Continuity and Differentiability - Continuity at a point

Definition of Continuity at a point
Symbolic representation of continuity at a point: $$f(a)$$
If a function is continuous at a point, then $$\lim_{{x \to a}} f(x)$$ exists
Graphical representation of continuity at a point
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Continuity in an interval

Definition of Continuity in an interval
Symbolic representation of continuity in an interval: $$f(x)$$
If a function is continuous in an interval, then it is continuous at every point in that interval
Graphical representation of continuity in an interval
Example: $$f(x) = \sin(x)$$ in the interval $$[-\pi, \pi]$$

Continuity and Differentiability - Discontinuity

Definition of Discontinuity
Types of Discontinuity: Removable Discontinuity, Jump Discontinuity, and Infinite Discontinuity
Symbolic representation of discontinuity
Graphical representation of discontinuity
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Types of Discontinuity

Removable Discontinuity: when there is a hole in the graph of a function at a certain point but can be filled
Jump Discontinuity: when there exist two one-sided limits at a certain point but they are not equal
Infinite Discontinuity: when the function approaches infinity or negative infinity at a certain point
Graphical representation of each type of discontinuity
Example: $$f(x) = \frac{{\sin(x)}}{{x}}$$ at $$x = 0$$

Continuity and Differentiability - Differentiability

Definition of Differentiability
Symbolic representation of differentiability: $$f’(x)$$
Differentiability implies continuity
Differentiability at a point vs Differentiability in an interval
Example: $$f(x) = \frac{{x^2 - 4}}{{x - 2}}$$ at $$x = 2$$

Continuity and Differentiability - Differentiation Rules

Power Rule: $$\frac{{d}}{{dx}}(x^n) = nx^{n-1}$$
Constant Multiple Rule: $$\frac{{d}}{{dx}}(cf(x)) = c \cdot \frac{{d}}{{dx}}(f(x))$$
Sum and Difference Rule: $$\frac{{d}}{{dx}}(f(x) \pm g(x)) = \frac{{d}}{{dx}}(f(x)) \pm \frac{{d}}{{dx}}(g(x))$$
Product Rule: $$\frac{{d}}{{dx}}(f(x) \cdot g(x)) = f’(x)g(x) + f(x)g’(x)$$
Quotient Rule: $$\frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{f’(x)g(x) - f(x)g’(x)}}{{g^2(x)}}$$