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# System prepended metadata

title: Objective

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# Objective

Going to attempt to add expand the functionality of Material-Math to test concepts from first year calculus such as taking derivatives and integrals. 

# Features

I want Material-Math to be able to test the following concepts from differential and integral calculus: 

**Differential Calculus**

* Power Rule
* Product Rule
* Chain Rule
* Quotient Rule

**Integral Calculus**

* Antipower Rule 
* U substition 
* Trig substition 
* Integration by parts 
* Partial fractions 

# Obstacles

Some of the biggest challenges that we will face include: 

* Figuring out a way to generate the problems. 
    * When generating functions to integrate we need to make sure that it actually has a solution. It is possible for a function not the have an elementary antiderivative. 
* Being able to solve the problems that we've generated. 
* Figure out how to classify problems as easy, medium, or hard. 

# Ideas

## Generating Problems 

We are targeting first year calculus students with these additional features. We can look at common types of problems from MAT135 material (textbook, midterm, exams) and try classifying them into different categories and generate problems of those type. 

#### Example on how you'd classify a type of problem (Integration by Parts)

| Type          | Example          | 
| ------------- |:-------------:| 
| Polynomial and exponential | $$\int (ax^2 + bx +c)e^{dx}$$ | 
| Polynomial and trigonometric     | $$\int (ax +b)sin(cx)$$      | 
| Inverse trigonometric| $$\int sin^{-1}(ax)$$    | 


We can make this list of problems in this category finite by putting constriants on the degree of the Polynomial and the constants. 

We also have the ability to make these problems more challenging by having students use multiple techniques. For example we can force students to use both u substitution and integration by parts by replacing the $x$ with a function $f(x)$ and multipliying by $f'(x)$. 

For example if we have 

$$\int sin^{-1}(x)$$

If we let $$f(x) = ln(x)$$ then $$f'(x) = \frac{1}{x}$$

And our new integral will be 

$$\int \frac{sin^{-1}(ln(x))}{x}$$. 



