# Calculus Problem Solving with Solution

Calculus problem solving and applications are presented with examples and their solutions.

## Calculus Problems with Solutions

- Calculus 1 Practice Question with detailed solutions.
- Maximize Volume of a Box. How to maximize the volume of a box using the first derivative of the volume.
- Use Derivatives to solve problems: Area Optimization. A problem to maximize (optimization) the area of a rectangle with a constant perimeter is presented.
- Solve Rate of Change Problems in Calculus. Calculus Rate of change problems and their solutions are presented.
- Optimization Problems for Calculus 1 with detailed solutions.
- Linear Least Squares Fitting. Use partial derivatives to find a linear fit for a given experimental data.
- Minimum Distance Problem. The first derivative is used to minimize distance traveled.
- Maximum Area of Rectangle - Problem with Solution. Maximize the area of a rectangle inscribed in a triangle using the first derivative. The problem and its solution are presented.
- Maximum Radius of Circle - Problem with Solution. Find the size of an angle of a right triangle so that the radius of the circle inscribed is maximum; for a constant hypotenuse.
- Find The Area of a Circle Using Integrals in Calculus.
- Find The Area of an Ellipse Using Calculus.
- Volume of a Spherical Cap using integrals.
- Find The Volume of a Sphere Using Calculus.
- Find The Volume of a Frustum Using Calculus.
- Find The Volume of a Square Pyramid Using Integrals.
- Maximum Area of Triangle - Problem with Solution. The first derivative is used to maximize the area of a triangle inscribed inside a circle.
- Maximum Area of Rectangle in a Right Triangle - Problem with Solution. Maximize the area of a rectangle inscribed in right triangle using the first derivative
- Maximize Power Delivered to Circuits. The first derivative is used to maximize the power delivered to a load in electronic circuits.
- Use Derivative to Find Quadratic Function. Use the first derivative to find the equation of a quadratic function given tangent lines to the graph of this function.
- Mean Value Theorem Problems. Problems, with detailed solutions, where the mean value theorem is used are presented.
- Rolle's Theorem Questions and Examples
- Use First Derivative to Minimize Area of Pyramid. The first derivative is used to minimize the surface area of a pyramid with a square base. A detailed solution to the problem is presented.
- Solve Tangent Lines Problems in Calculus. Tangent lines problems and their solutions are presented.
- Use Derivatives to solve problems: Distance-time Optimization. A problem to minimize (optimization) the time taken to walk from one point to another is presented.
- Minimum, Maximum, First and Second Derivatives. A tutorial on how to use calculus theorems using first and second derivatives to determine whether a function has a relative maximum or minimum or neither at a given point.
- First, Second Derivatives and Graphs Of Functions. A tutorial on how to use the first and second derivatives, in calculus, to graph functions.

## Application of Differentiation and Derivatives

- Absolute Minimum and Maximum of a Function, examples with detailed solutions and graphical interpretations.
- Newton's Method to Find Zeros of a Function. Newton's method is an example of how differentiation is used to find zeros of functions and solve equations numerically. Examples with detailed solutions on how to use Newton s method are presented.
- Linear Approximation of Functions. Linear approximation is another example of how differentiation is used to approximate functions by linear ones close to a given point. Examples with detailed solutions on linear approximations are presented.
- Find Critical Numbers of Functions. Tutorial on how to find the critical numbers of a function. Several examples with detailed solutions and exercises with answers.
- Derivative, Maximum, Minimum of Quadratic Functions. Differentiation is used to analyze the properties such as intervals of increase, decrease, local maximum, local minimum of quadratic functions. Examples with solutions and exercises with answers.
- Determine the Concavity of Quadratic Functions. Examples with solutions and exercises with answers.
- Use Derivative to Show That arcsin(x) + arccos(x) = pi/2.

## Applications of Integrals

- Find Area Under Curve. How to find the area under (and between) curves using definite integrals; tutorials, with examples and detailed solutions are presented.
- Find Area Between Curves. How to find the area between curves using definite integrals.
- Length of a Curve. Examples with detailed solutions.
- Calculate The Area in Polar Coordinates Curve. Examples with detailed solutions.
- Area of an Ellipse in Polar Coordinates .
- Find The Volume of a Solid of Revolution. How to find the volume of a solid of revolution generated by revolving a region bounded by the graph of a function around one of the axes using definite integrals?
- Volume by Cylindrical Shells Method. Find the volume of a solid of revolution generated by revolving a region bounded by the graph of a function around one of the axes using cylindrical shells.

## Parametric Equations and their Applications

- Dervative of Parametric Equations and Applications. Examples and questions with solutions.

## Applications of Multivariable Functions

- Maxima and Minima of Functions of Two Variables. Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points.
- Linear Least Squares Fitting. Use partial derivatives to find a linear fit for a given experimental data.
- Optimization Problems with Functions of Two Variables. Several optimization problems are solved and detailed solutions are presented. These problems involve optimizing functions in two variables using first and second order partial derivatives..