Derivatives - Intermediate Value Theorem (IVT)

• The Intermediate Value Theorem (IVT) is a fundamental result in calculus.
• It is used to prove the existence of roots or solutions to equations.
• The theorem states that if a function is continuous on a closed interval [a, b], and we have two points y1 and y2 such that y1 < c < y2, then there exists at least one point c in the open interval (a, b) where the function takes the value c. Example: Let’s consider the function f(x) = x^2 - 4x + 3 on the interval [1, 3].
• The value of f(1) is -2 and f(3) is 0.
• Since 0 is between -2 and 0, the Intermediate Value Theorem guarantees the existence of a root between x = 1 and x = 3. Equation: f(x) = x^2 - 4x + 3 = 0

Derivatives - Rolle’s Theorem

• Rolle’s theorem is a special case of the Mean Value Theorem (MVT).
• It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and the function takes the same value at both endpoints, then there exists at least one point c in the interval (a, b) where the derivative is zero. Example: Let’s consider the function f(x) = x^2 - 3x on the interval [1, 2].
• The value of f(1) is -2 and f(2) is 0.
• The derivative of f(x) is 2x - 3.
• By Rolle’s Theorem, there exists at least one point c in the interval (1, 2) where the derivative is zero. Equation: f’(x) = 2x - 3 = 0

Derivatives - Increasing and Decreasing Functions

• A function is said to be increasing if its derivative is positive on its entire domain.
• A function is said to be decreasing if its derivative is negative on its entire domain.
• Local maxima and minima occur where the function changes from increasing to decreasing or vice versa.
• Inflection points occur where the function changes concavity. Example: Let’s consider the function f(x) = x^3 - 6x^2 + 9x on its domain.
• The derivative of f(x) is 3x^2 - 12x + 9.
• To determine intervals of increase and decrease, solve f’(x) > 0 and f’(x) < 0.
• Increase: (-∞, 1), (3, ∞)
• Decrease: (1, 3)
• The function has a local maximum at x = 1 and a local minimum at x = 3. Equations: f’(x) > 0 and f’(x) < 0

Derivatives - Concavity and Convexity

• A function is said to be concave up (or convex) if its second derivative is positive on its entire domain.
• A function is said to be concave down (or concave) if its second derivative is negative on its entire domain.
• Inflection points occur where the concavity changes. Example: Let’s consider the function f(x) = x^4 - 4x^2 on its domain.
• The second derivative of f(x) is 12x^2 - 8.
• To determine concavity, solve f’’(x) > 0 and f’’(x) < 0.
• Concave up: (-∞, -√2), (√2, ∞)
• Concave down: (-√2, √2)
• The function has an inflection point at x = -√2 and x = √2. Equations: f’’(x) > 0 and f’’(x) < 0

Derivatives - Limits at Infinity

• As x approaches infinity or negative infinity, functions may have limits.
• To find the limit, we examine the leading terms of the function.
• There are three cases: (1) Polynomials, (2) Rational Functions, and (3) Exponential/Logarithmic Functions. Example 1: Polynomials
• Let’s consider the function f(x) = x^3 - 2x + 1.
• As x approaches infinity, the leading term is x^3, so the limit is infinity.
• As x approaches negative infinity, the leading term is -x^3, so the limit is negative infinity. Example 2: Rational Functions
• Let’s consider the function f(x) = (2x + 3) / (x - 1).
• As x approaches infinity, the leading terms are 2x/x = 2, so the limit is 2.
• As x approaches negative infinity, the leading terms are -2x/x = -2, so the limit is -2. Example 3: Exponential/Logarithmic Functions
• Let’s consider the function f(x) = e^x / ln(x).
• As x approaches infinity, both the numerator and denominator approach infinity, so the limit is indeterminate.
• As x approaches negative infinity, both the numerator and denominator approach zero, so the limit is indeterminate.

Derivatives - Curve Sketching

• Curve sketching involves analyzing the characteristics of a function to create an accurate graph.
• Steps for curve sketching: (1) Domain and Range, (2) Intercepts, (3) Symmetry, (4) Intervals of Increase/Decrease, (5) Local Maxima/Minima, (6) Concavity, and (7) Asymptotes.
• We use derivatives and their properties to determine these characteristics. Example: Let’s consider the function f(x) = x^4 - 4x^2.

1. Domain: All real numbers, Range: [0, ∞)

2. Intercepts: f(0) = 0, x-intercepts: x = -2, x = 2

3. Symmetry: Even function, symmetric about the y-axis

4. Increase/Decrease: Increasing on (-∞, -2) and (2, ∞); Decreasing on (-2, 2)

5. Local Maxima/Minima: Local maxima at x = -2 and local minimum at x = 2

6. Concavity: Concave up on (-∞, -√2) and (√2, ∞); Concave down on (-√2, √2)

7. Asymptotes: No vertical asymptotes; Horizontal asymptote at y = 0

Derivatives - Optimization

• Optimization problems involve finding the maximum or minimum value of a function within a given domain or constraint.
• Steps for optimization: (1) Define variables and constraints, (2) Write the function to be optimized, (3) Find critical points, (4) Evaluate endpoints or boundaries, and (5) Determine the maximum or minimum value. Example: Let’s optimize the area of a rectangle with a fixed perimeter of 20 units.

1. Variables: Length (L) and Width (W), Constraint: 2L + 2W = 20

2. Function to be optimized: Area (A) = L * W

3. Constraint: W = 10 - L

4. Substitute W in terms of L into the function: A = L(10 - L) = 10L - L^2

5. Find critical points: Set A’ = 0, A’’ < 0 to find the maximum point.
   • A’ = 10 - 2L = 0, L = 5

6. Evaluate endpoints: L = 0 and L = 10 (boundaries)

7. Determine the maximum value: A(0) = A(10) = 0, A(5) = 25 (maximum) The maximum area of the rectangle is 25 square units when L = 5 and W = 10 - L = 5.

Derivatives - Related Rates

• Related rates involve finding the rate of change of one quantity with respect to another when they are connected through an equation.
• Steps for related rates: (1) Identify the known and unknown variables, (2) Write an equation connecting the variables, (3) Differentiate both sides of the equation with respect to time, (4) Substitute the known values, and (5) Solve for the unknown rate of change. Example: A balloon is being inflated at a rate of 2 cm³/s. The radius is increasing at a constant rate of 0.5 cm/s. Find the rate of change of the volume when the radius is 10 cm.

1. Known variables: dV/dt = 2 cm³/s, dr/dt = 0.5 cm/s, r = 10 cm

2. Equation: V = (4/3)πr³ (volume of a sphere)

3. Differentiate both sides with respect to time: dV/dt = 4πr²(dr/dt)

4. Substitute the known values: dV/dt = 4π(10)²(0.5)

5. Solve for dV/dt: dV/dt = 100π cm³/s The rate of change of the volume when the radius is 10 cm is 100π cm³/s.

Derivatives - Mean Value Theorem (MVT)

• The Mean Value Theorem (MVT) is a fundamental result in calculus.
• It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) where the instantaneous rate of change is equal to the average rate of change. Example: Let’s consider the function f(x) = x^2 - 2x on the interval [0, 2].
• The average rate of change = (f(2) - f(0)) / (2 - 0) = (0 - 0) / 2 = 0
• The derivative of f(x) is 2x - 2.
• By MVT, there exists at least one point c in the interval (0, 2) where the derivative is equal to the average rate of change, which is 0. Equation: f’(c) = 2c - 2 = 0

Derivatives - Intermediate Value Theorem (IVT)

• The Intermediate Value Theorem (IVT) is a fundamental result in calculus.
• It is used to prove the existence of roots or solutions to equations.
• The theorem states that if a function is continuous on a closed interval [a, b], and we have two points y1 and y2 such that y1 < c < y2, then there exists at least one point c in the open interval (a, b) where the function takes the value c.
• The IVT is also applicable to other mathematical objects, such as sequences.
• The IVT is often used to determine if a function has a zero or a root. Example: Let’s consider the function f(x) = x^2 - 4x + 3 on the interval [1, 3].
• The value of f(1) is -2 and f(3) is 0.
• Since 0 is between -2 and 0, the Intermediate Value Theorem guarantees the existence of a root between x = 1 and x = 3.
• The theorem does not specify how many roots exist, but it guarantees at least one root. Equation: f(x) = x^2 - 4x + 3 = 0

Derivatives - Rolle’s Theorem

• Rolle’s theorem is a special case of the Mean Value Theorem (MVT).
• It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and the function takes the same value at both endpoints, then there exists at least one point c in the interval (a, b) where the derivative is zero.
• Another way to state Rolle’s theorem is that between any two points on the graph of a continuous differentiable function, there must be at least one point where the tangent line is horizontal.
• The theorem is named after the French mathematician Michel Rolle who first proved it. Example: Let’s consider the function f(x) = x^2 on the interval [0, 1].
• The value of f(0) is 0 and f(1) is 1.
• The derivative of f(x) is 2x.
• By Rolle’s Theorem, there exists at least one point c in the interval (0, 1) where the derivative is zero.
• In this case, the derivative is zero when x = 0.5. Equation: f’(x) = 2x = 0

Derivatives - Increasing and Decreasing Functions

• A function is said to be increasing if its derivative is positive on its entire domain.
• A function is said to be decreasing if its derivative is negative on its entire domain.
• Intuitively, an increasing function has a positive slope while a decreasing function has a negative slope.
• The first derivative test can be used to determine intervals of increase and decrease. Example: Let’s consider the function f(x) = x^2 - 2 on its domain.
• The derivative of f(x) is 2x.
• To determine intervals of increase and decrease, solve f’(x) > 0 and f’(x) < 0.
• Increase: (0, ∞)
• Decrease: (-∞, 0)
• The function is increasing on the interval (0, ∞) and decreasing on the interval (-∞, 0). Equations: f’(x) > 0 and f’(x) < 0

Derivatives - Concavity and Convexity

• A function is said to be concave up (or convex) if its second derivative is positive on its entire domain.
• A function is said to be concave down (or concave) if its second derivative is negative on its entire domain.
• Intuitively, a concave up function is shaped like a ‘U’ while a concave down function is shaped like a ’n’.
• The second derivative test can be used to determine intervals of concavity. Example: Let’s consider the function f(x) = x^3 - 6x^2 + 9x on its domain.
• The second derivative of f(x) is 6x - 12.
• To determine concavity, solve f’’(x) > 0 and f’’(x) < 0.
• Concave up: (2, ∞)
• Concave down: (-∞, 2)
• The function is concave up on the interval (2, ∞) and concave down on the interval (-∞, 2). Equations: f’’(x) > 0 and f’’(x) < 0

Derivatives - Limits at Infinity

• As x approaches infinity or negative infinity, functions may have limits.
• To find the limit, we examine the leading terms of the function.
• There are three cases: (1) Polynomials, (2) Rational Functions, and (3) Exponential/Logarithmic Functions.
• The limit can be positive infinity, negative infinity, a finite value, or it may not exist. Example 1: Polynomials
• Let’s consider the function f(x) = x^3 - 2x + 1.
• As x approaches infinity, the leading term is x^3, so the limit is infinity.
• As x approaches negative infinity, the leading term is -x^3, so the limit is negative infinity. Example 2: Rational Functions
• Let’s consider the function f(x) = (2x + 3) / (x - 1).
• As x approaches infinity, the leading terms are 2x/x = 2, so the limit is 2.
• As x approaches negative infinity, the leading terms are -2x/x = -2, so the limit is -2. Example 3: Exponential/Logarithmic Functions
• Let’s consider the function f(x) = e^x / ln(x).
• As x approaches infinity, both the numerator and denominator approach infinity, so the limit is indeterminate.
• As x approaches negative infinity, both the numerator and denominator approach zero, so the limit is indeterminate.

Derivatives - Curve Sketching

• Curve sketching involves analyzing the characteristics of a function to create an accurate graph.
• Steps for curve sketching: (1) Domain and Range, (2) Intercepts, (3) Symmetry, (4) Intervals of Increase/Decrease, (5) Local Maxima/Minima, (6) Concavity, and (7) Asymptotes.
• We use derivatives and their properties to determine these characteristics.
• Curve sketching helps us visualize the behavior of a function and understand its key features. Example: Let’s consider the function f(x) = x^4 - 4x^2.

1. Domain: All real numbers, Range: [0, ∞)

2. Intercepts: f(0) = 0, x-intercepts: x = -2, x = 2

3. Symmetry: Even function, symmetric about the y-axis

4. Increase/Decrease: Increasing on (-∞, -2) and (2, ∞); Decreasing on (-2, 2)

5. Local Maxima/Minima: Local maxima at x = -2 and local minimum at x = 2

6. Concavity: Concave up on (-∞, -√2) and (√2, ∞); Concave down on (-√2, √2)

7. Asymptotes: No vertical asymptotes; Horizontal asymptote at y = 0

Derivatives - Optimization

• Optimization problems involve finding the maximum or minimum value of a function within a given domain or constraint.
• Steps for optimization: (1) Define variables and constraints, (2)