Unit 3: Application of Derivatives

1. Rolle’s Theorem

Rolle's Theorem provides the conditions under which a differentiable function attains a horizontal tangent line (derivative equals zero).

Statement

If a function satisfies the following three conditions:

1. Continuity: is continuous on the closed interval .
2. Differentiability: is differentiable on the open interval .
3. Equality: .

Then, there exists at least one real number in the interval such that:

Geometric Interpretation

Geometrically, Rolle’s Theorem asserts that if a smooth curve connects two points at the same height (-level), there must be at least one point on the curve between them where the tangent line is horizontal (parallel to the x-axis).

Algebraic Interpretation

Between any two roots of a differentiable function , there lies at least one root of its derivative .

2. Mean Value Theorems

The Mean Value Theorem (MVT), often referred to as Lagrange's Mean Value Theorem, is a generalization of Rolle's Theorem.

Lagrange’s Mean Value Theorem (LMVT)

Statement:
If a function satisfies:

1. Continuity: is continuous on the closed interval .
2. Differentiability: is differentiable on the open interval .

Then, there exists at least one point such that:

Geometric Interpretation:
There is at least one point on the curve where the tangent line is parallel to the secant line connecting the endpoints and .

Key Relationship:
If , the numerator becomes 0, and LMVT reduces to Rolle's Theorem ().

Cauchy’s Mean Value Theorem

A generalization of LMVT involving two functions.

Statement:
If functions and are continuous on and differentiable on , and for all , then there exists a such that:

3. Taylor’s Theorem with Remainders

Taylor's theorem allows the approximation of a -times differentiable function around a specific point by a polynomial.

Statement

If a function possesses continuous derivatives up to order on and exists on , then can be expanded about as:

Where is the Remainder term (or error term) after terms.

Forms of the Remainder ()

1. Lagrange’s Form of Remainder:
   

   
   Where . This form is most commonly used for estimating the maximum error of the approximation.

2. Cauchy’s Form of Remainder:
   

   
   Where .

4. Maclaurin’s Theorem with Remainders

Maclaurin’s Theorem is a special case of Taylor’s Theorem where the expansion is centered at .

Statement

If is defined in an interval containing 0 and its derivatives exist up to order , then for any in that interval:

Important Maclaurin Series Expansions

Engineers frequently use infinite Maclaurin series (where as ).

1. Exponential:
   
2. Sine:
   
3. Cosine:
   
4. Logarithm:
   

5. Indeterminate Forms

When evaluating limits, substituting the limit value sometimes results in undefined expressions known as indeterminate forms.

The 7 Indeterminate Forms

1. Standard Forms:
2. Product Form:
3. Difference Form:
4. Exponential Forms:

L'Hospital's Rule (L'Hôpital's Rule)

Applicability: Strictly for forms or .

Statement:
If results in or , then:

Handling Non-Standard Forms

1. Form :
   Rewrite as to convert to or .

2. Form :
   Usually involves fractions. Combine terms using a common denominator or use rationalization to convert to a ratio.

3. Forms :
   Let .
   Take the natural log () of both sides:
   

   
   This converts the limit to the form . Solve for , then exponentiate to find .

6. Maxima and Minima

This section deals with finding the optimum points (peaks and valleys) of a function.

Definitions

• Local Maximum: for all in the immediate neighborhood of .
• Local Minimum: for all in the immediate neighborhood of .
• Stationary (Critical) Points: Points where or does not exist.

First Derivative Test

1. Find .
2. Solve to find critical points .
3. Check the sign of on the left and right of :
   • Max: Sign changes from positive to negative.
   • Min: Sign changes from negative to positive.
   • Inflection: No sign change (neither max nor min).

Second Derivative Test (Preferred Method)

1. Find and solve for critical points .
2. Find the second derivative .
3. Substitute into :
   • If : The graph is concave down Local Maximum at .
   • If : The graph is concave up Local Minimum at .
   • If : The test fails (use the First Derivative Test or higher-order derivatives).

Application to Optimization Problems

1. Identify Variables: Assign symbols to the quantities (e.g., radius , height ).
2. Constraint Equation: Identify the fixed condition given in the problem (e.g., Volume ). Use this to express one variable in terms of the other.
3. Objective Function: Write the formula for the quantity to be maximized or minimized (e.g., Surface Area ).
4. Differentiate: Substitute the constraint into the objective function so it relies on a single variable, then differentiate and set to zero.
5. Verify: Use the second derivative test to ensure the solution is indeed a maximum or minimum.