How to write Algorithm?

Problem 1: Solve a Quadratic Equation

1. Begin

2. Input coefficients a, b, and c

3. Calculate discriminant (b^2 - 4ac)

4. Check if discriminant is negative

5. If yes, display "No real roots," exit

6. If no, calculate roots using quadratic formula

7. Display roots

8. End

Problem 2: Evaluate a Definite Integral

1. Begin

2. Input function f(x) and integration limits a, b

3. Divide the interval [a, b] into n subintervals

4. Choose sample points in each subinterval

5. Calculate function values at sample points

6. Multiply values by subinterval width

7. Sum these products to approximate integral

8. Display result

9. End

Problem 3: Find the Eigenvalues of a Matrix

1. Begin

2. Input square matrix A

3. Calculate determinant |A - λI|

4. Solve characteristic equation for eigenvalues λ

5. Display eigenvalues

6. End

Problem 4: Solve a System of Linear Equations

1. Begin

2. Input coefficients matrix A and constants vector B

3. Form augmented matrix [A|B]

4. Apply Gaussian elimination to get row-echelon form

5. Back-substitute to find solution

6. Display solution vector

7. End

Problem 5: Determine the Taylor Series Expansion

1. Begin

2. Input function f(x) and expansion point a

3. Calculate nth derivative of f(x)

4. Evaluate derivatives at x=a

5. Write Taylor series using derivatives

6. Display Taylor series

7. End

Problem 6: Optimize a Function

1. Begin

2. Input function f(x), find its derivative f'(x)

3. Set f'(x) = 0 and solve for critical points

4. Test critical points and endpoints for extrema

5. Display optimized solution

6. End

Problem 7: Solve a Differential Equation

1. Begin

2. Input differential equation and initial conditions

3. Determine the order and type of the equation

4. Choose an appropriate method (e.g., separation of variables)

5. Integrate to find the general solution

6. Apply initial conditions to find particular solution

7. Display solution

8. End

Problem 8: Perform Matrix Multiplication

1. Begin

2. Input matrices A and B

3. Check if dimensions allow multiplication

4. Multiply corresponding elements to form product matrix

5. Display product matrix

6. End

Problem 9: Find the Limit of a Function

1. Begin

2. Input function f(x) and limit point c

3. Analyze behavior of f(x) as x approaches c

4. Apply limit rules or L'Hôpital's rule if necessary

5. Display limit value

6. End

Problem 10: Solve a Trigonometric Identity

1. Begin

2. Input trigonometric expression

3. Use trigonometric identities to simplify

4. Apply algebraic manipulations