CALCULUS PRESENTATION.pptx.ecnomic and finance course

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1. CALCULUS PRESENTATION

2. DIFFERENTIATION & DERIVIATION

3. Presentation by Collaborative Crews Group Members: Ali Haider Muhammad Zaeem Arooj Zahid Minahil Sherazi Muhammad yousuf Abbas Zaidi

4. Introduction A derivative is a fundamental concept in calculus and mathematical analysis that measures how a function changes as its input changes. It provides a way to describe the rate of change or slope of a function at a given point.

5. Objectives: Finding Instantaneous Rates of Change : Derivation helps in determining how a quantity changes at a particular point in time or space. For example, velocity is the derivative of position with respect to time. Solving Real-World Problems : Derivation allows modeling real-world scenarios such as growth rates, physical changes, and optimization problems. Understanding Function Behavior : It helps analyze how a function behaves, such as identifying increasing or decreasing intervals, concavity, and points of inflection.

6. 1. Definition of a Derivative: Mathematically, the derivative of a function f(x) at a point x is define as: (x)= This formula gives the slope of the tangent line to the function at a given point.  

7. 2. Geometric Interpretation: The derivative at a point corresponds to the slope of the tangent line to the graph of f(x) at that point. If f′(a)> 0, the function is increasing at x=a. If f′(a)< 0, the function is decreasing at x=a. If f′(a)= 0, the point may be a maximum, minimum, or inflection point.

8. 3. Notation: f′(x): Prime notation. Leibniz notation, emphasizing the ratio of changes. f ˙​(x): Newton's notation, often used in physics. Df (x ): Operator notation.  

9. 4. Applications: Derivatives are widely used in various fields: Physics : To determine velocity and acceleration. Economics : To analyze marginal costs and profits. Engineering : To optimize designs and systems. Biology : To model population growth rates.

10. 5. Basic Rules of Differentiation: 1 . Constant Rule The derivative of a constant is zero . where c is a constant . 2 . Power Rule For a function f(x) = , the derivative: Where, n is any real number. Chain Rule: For a composite function f[g(x)]:  

11. Product Rule For product of two functions u(x) and v(x). Quotient Rule For division of two functions u(x) and v(x): =  

12. Product Rule: The Product Rule is a fundamental rule in calculus for finding the derivative of the product of two functions . Formula : If u(x) and v(x) are two differentiable functions, the derivative of their product is given by: Explanation: u(x): The first function. v(x): The second function. u ′(x): The derivative of u(x). v′( x): The derivative of v(x).  

13. Example: Let Here, U(x) = 2 and v(x)= sin(x) 1. 2. Using the product rule:  

14. QUOTIENT RULE : The quotient rule is used in calculus to find the derivative of a function that is the division of two differentiable functions, say f(x)= where u(x) and v(x) are differentiable. Quotient Rule Formula: = where: u(x) is the numerator v(x) is the denominator is the derivative of v(x) is the derivative of u(x)  

15. Example: Find the derivatives of f(x) = Here: u(x) = and v(x) = x + 1 = 2x and (x) = 1 Using the formula: Simplify: = =  

16. Conclusions: Here are some basic conclusions of derivation: P redictive power: Derivative provide a mathematical way to predict how much change in one quantity ( e.g ; time , cost) will effect other. Optimization: It identify maximum and minimum values, which is essential resource management , logistic power and even personal budgeting. Decision making : By analyzing rate of change derivative allow businesses and individuals to make informed decisions, whether its about investing , hiring or, scheduling. Problem solving: Derivative simplify complex real world problem into solvable equation.

17. Thank You