Derivatives - Application Of Rolle Theorem & Mvt

Derivatives - Application of Rolle Theorem & MVT

Objectives

Understand the concepts of Rolle’s Theorem and the Mean Value Theorem.
Apply Rolle’s Theorem to find solutions to specific problems.
Apply the Mean Value Theorem to solve problems involving rates of change.
Solve real-life problems using these concepts.

Review: Rolle’s Theorem

Statement: Let f be a function that satisfies the following conditions:
- f is continuous on the closed interval [a, b].
- f is differentiable on the open interval (a, b).
- f(a) = f(b)
Conclusion: There exists at least one number c in (a, b) such that f’(c) = 0.

Example: Applying Rolle’s Theorem

Problem: Show that the function f(x) = x^3 - 5x + 1 has a root on the interval [1, 2].
Solution:
- f is a continuous function on [1, 2].
- f is differentiable on (1, 2).
- f(1) = 1^3 - 5(1) + 1 = -3
- f(2) = 2^3 - 5(2) + 1 = -3
Conclusion: By Rolle’s Theorem, there exists at least one number c in (1, 2) such that f’(c) = 0.

Review: Mean Value Theorem (MVT)

Statement: Let f be a function that satisfies the following conditions:
- f is continuous on the closed interval [a, b].
- f is differentiable on the open interval (a, b).
Conclusion: There exists at least one number c in (a, b) such that f’(c) = (f(b) - f(a))/(b - a).

Example: Applying MVT

Problem: Find the value of c guaranteed by the MVT for the function f(x) = x^2 - 4x + 3 on the interval [1, 3].
Solution:
- f is a continuous function on [1, 3].
- f is differentiable on (1, 3).
Conclusion: By the Mean Value Theorem, there exists at least one number c in (1, 3) such that f’(c) = (f(3) - f(1))/(3 - 1).

Applications of Rolle’s Theorem

Rolle’s Theorem can be used to prove other important theorems in calculus.
It is often used to determine whether a function has roots on a given interval.
It can be used to prove the existence of local extrema for a function.

Applications of the MVT

The MVT is used in many areas of calculus, including optimization problems.
It guarantees the existence of a tangent line with a specific slope on a given interval.
It is used to prove the derivative rules, such as the power rule and product rule.
It can be used to prove the existence of limits of certain functions.

Example: Applications of the MVT

Problem:
- A car travels a distance of 100 miles in 2 hours. Show that at some point during the trip, the car must have been traveling at exactly 50 miles per hour.
- The speed of the car is given by the function s(t), where t is the time in hours.
Solution:
- We know that s(t) is continuous on [0, 2] and differentiable on (0, 2).
- By the MVT, there exists a number c in (0, 2) such that s’(c) = (s(2) - s(0))/(2 - 0).

Summary

Rolle’s Theorem and the Mean Value Theorem are important concepts in calculus.
Rolle’s Theorem guarantees the existence of a derivative equal to zero on a given interval.
The Mean Value Theorem guarantees the existence of a derivative equal to the average rate of change on a given interval.
These theorems have various applications in calculus and are used to prove many important results.

Application of Rolle’s Theorem: Maximum and Minimum Values

Rolle’s Theorem can be used to find the maximum and minimum values of a function.
If a function satisfies the conditions of Rolle’s Theorem on an interval [a, b], then we can conclude that it must have a maximum or minimum value at some point within that interval.
This is because if the function is continuous and differentiable, and its derivative is zero at some point, then the function must either be increasing or decreasing on either side of that point.

Finding Maximum and Minimum Values: Example

Problem: Find the maximum and minimum values of the function f(x) = x^2 - 4x on the interval [0, 4].
Solution:
- We can see that f(x) is a quadratic function, which means it is continuous and differentiable on its entire domain.
- To find the maximum and minimum values, we need to find the critical points of the function by setting its derivative equal to zero.
- Differentiating f(x) with respect to x, we get f’(x) = 2x - 4.
- Setting f’(x) = 0, we get 2x - 4 = 0, which gives x = 2 as the only critical point.
- Plugging x = 2 into the original function, we get f(2) = 2^2 - 4(2) = -4.
- Therefore, the maximum value of f(x) on the interval [0, 4] is -4, and there is no minimum value.

Application of MVT: Average Rate of Change

The Mean Value Theorem can be used to find the average rate of change of a function on a given interval.
The average rate of change is the difference in the values of the function divided by the difference in the input values.
The MVT guarantees that there exists at least one point within the interval where the instantaneous rate of change is equal to the average rate of change.

Finding Average Rate of Change: Example

Problem: Find the average rate of change of the function f(x) = x^3 on the interval [0, 2].
Solution:
- To find the average rate of change, we need to calculate the difference in the values of the function and divide it by the difference in the input values.
- f(0) = 0^3 = 0 and f(2) = 2^3 = 8.
- The difference in the values of the function is 8 - 0 = 8.
- The difference in the input values is 2 - 0 = 2.
- Therefore, the average rate of change of f(x) on the interval [0, 2] is 8/2 = 4.

Application of MVT: Tangent Line

The Mean Value Theorem can be used to prove the existence of a tangent line to a curve with a specific slope.
If a function satisfies the conditions of the MVT on an interval, then there exists at least one point within that interval where the tangent line to the curve is parallel to the secant line connecting the endpoints of the interval.

Finding the Equation of the Tangent Line: Example

Problem: Find the equation of the tangent line to the curve y = x^2 at the point (3, 9).
Solution:
- We know that the slope of the tangent line is equal to the derivative of the function at the point.
- Differentiating y = x^2 with respect to x, we get dy/dx = 2x.
- Evaluating the derivative at x = 3, we get dy/dx = 2(3) = 6.
- Therefore, the equation of the tangent line at the point (3, 9) is y - 9 = 6(x - 3).

Applications of the MVT: Optimization Problems

The Mean Value Theorem can be used to solve optimization problems.
Optimization problems involve finding the maximum or minimum value of a quantity subject to certain constraints.
By applying the MVT to the appropriate function, we can find the values that optimize the quantity.

Solving Optimization Problems: Example

Problem: Find the dimensions of a rectangle with maximum area if its perimeter is fixed at 12 units.
Solution:
- Let’s assume the length of the rectangle is x units and the width is y units.
- The area of the rectangle is given by A = xy.
- The perimeter of the rectangle is given by P = 2x + 2y.
- From the problem statement, we have 2x + 2y = 12, which gives x + y = 6.
- We need to optimize A = xy given the constraint x + y = 6.
- We can rewrite x = 6 - y and substitute it into the formula for A.
- Now, we have A = (6 - y)y = 6y - y^2.
- Differentiating A with respect to y, we get dA/dy = 6 - 2y.
- Setting dA/dy = 0, we get y = 3.
- Plugging y = 3 into x + y = 6, we get x = 3.
- Therefore, the dimensions of the rectangle with maximum area are 3 units by 3 units.

Real-Life Applications of Calculus

Calculus has numerous real-life applications, including:
- Calculating rates of change in physics, such as velocity and acceleration.
- Modeling population growth and decay in biology and sociology.
- Optimizing production and profit in economics and business.
- Predicting weather patterns using differential equations in meteorology.

Summary

Rolle’s Theorem and the Mean Value Theorem are powerful tools in calculus.
Rolle’s Theorem guarantees the existence of a zero derivative on a given interval, while the MVT guarantees the existence of a derivative equal to the average rate of change.
These theorems can be used to find maximum and minimum values, solve optimization problems, and prove important calculus concepts.
Calculus has wide-ranging applications in various disciplines and is a fundamental tool for understanding the world around us.

Derivatives - Application of Rolle Theorem & MVT

Slide 21

Real-Life Applications of Calculus (Continued)
- Calculating rates of change in physics, such as velocity and acceleration.
- Modeling population growth and decay in biology and sociology.
- Optimizing production and profit in economics and business.
- Predicting weather patterns using differential equations in meteorology.

Slide 22

Summary
- Rolle’s Theorem and the Mean Value Theorem are powerful tools in calculus.
- Rolle’s Theorem guarantees the existence of a zero derivative on a given interval, while the MVT guarantees the existence of a derivative equal to the average rate of change.
- These theorems can be used to find maximum and minimum values, solve optimization problems, and prove important calculus concepts.
- Calculus has wide-ranging applications in various disciplines and is a fundamental tool for understanding the world around us.

Slide 23

Example: Applying Rolle’s Theorem
- Problem: Show that the function f(x) = x^3 - 5x + 1 has a root on the interval [1, 2].
- Solution:
  - f is a continuous function on [1, 2].
  - f is differentiable on (1, 2).
  - f(1) = 1^3 - 5(1) + 1 = -3
  - f(2) = 2^3 - 5(2) + 1 = -3
- Conclusion: By Rolle’s Theorem, there exists at least one number c in (1, 2) such that f’(c) = 0.

Slide 24

Example: Applying MVT
- Problem: Find the value of c guaranteed by the MVT for the function f(x) = x^2 - 4x + 3 on the interval [1, 3].
- Solution:
  - f is a continuous function on [1, 3].
  - f is differentiable on (1, 3).
- Conclusion: By the Mean Value Theorem, there exists at least one number c in (1, 3) such that f’(c) = (f(3) - f(1))/(3 - 1).

Slide 25

Application of Rolle’s Theorem: Maximum and Minimum Values
- Rolle’s Theorem can be used to find the maximum and minimum values of a function.
- If a function satisfies the conditions of Rolle’s Theorem on an interval [a, b], then we can conclude that it must have a maximum or minimum value at some point within that interval.
- This is because if the function is continuous and differentiable, and its derivative is zero at some point, then the function must either be increasing or decreasing on either side of that point.

Slide 26

Finding Maximum and Minimum Values: Example
- Problem: Find the maximum and minimum values of the function f(x) = x^2 - 4x on the interval [0, 4].
- Solution:
  - We can see that f(x) is a quadratic function, which means it is continuous and differentiable on its entire domain.
  - To find the maximum and minimum values, we need to find the critical points of the function by setting its derivative equal to zero.
  - Differentiating f(x) with respect to x, we get f’(x) = 2x - 4.
  - Setting f’(x) = 0, we get 2x - 4 = 0, which gives x = 2 as the only critical point.
  - Plugging x = 2 into the original function, we get f(2) = 2^2 - 4(2) = -4.
  - Therefore, the maximum value of f(x) on the interval [0, 4] is -4, and there is no minimum value.

Slide 27

Application of the MVT: Average Rate of Change
- The Mean Value Theorem can be used to find the average rate of change of a function on a given interval.
- The average rate of change is the difference in the values of the function divided by the difference in the input values.
- The MVT guarantees that there exists at least one point within the interval where the instantaneous rate of change is equal to the average rate of change.

Slide 28

Finding Average Rate of Change: Example
- Problem: Find the average rate of change of the function f(x) = x^3 on the interval [0, 2].
- Solution:
  - To find the average rate of change, we need to calculate the difference in the values of the function and divide it by the difference in the input values.
  - f(0) = 0^3 = 0 and f(2) = 2^3 = 8.
  - The difference in the values of the function is 8 - 0 = 8.
  - The difference in the input values is 2 - 0 = 2.
  - Therefore, the average rate of change of f(x) on the interval [0, 2] is 8/2 = 4.

Slide 29

Application of the MVT: Tangent Line
- The Mean Value Theorem can be used to prove the existence of a tangent line to a curve with a specific slope.
- If a function satisfies the conditions of the MVT on an interval, then there exists at least one point within that interval where the tangent line to the curve is parallel to the secant line connecting the endpoints of the interval.

Slide 30

Finding the Equation of the Tangent Line: Example
- Problem: Find the equation of the tangent line to the curve y = x^2 at the point (3, 9).
- Solution:
  - We know that the slope of the tangent line is equal to the derivative of the function at the point.
  - Differentiating y = x^2 with respect to x, we get dy/dx = 2x.
  - Evaluating the derivative at x = 3, we get dy/dx = 2(3) = 6.
  - Therefore, the equation of the tangent line at the point (3, 9) is y - 9 = 6(x - 3).