The fundamental theorem of calculus

The fundamental theorem of calculus exercise appears under the Integral calculus Math Mission. This exercise shows the connection between derivative calculus and integral calculus.

Types of Problems
There are three types of problems in this exercise:



1. Find the derivative of the integral: The student is asked to find the derivative of a given integral using the fundamental theorem of calculus.

2. Find the tangent line from the graph of an defined integral: The student is asked to find the tangent line in slope-intercept form or point-slope form using the graph of the integral.

3. Define the integral when it is decreasing/increasing on the interval(s): The student is asked to define when the integral function is decreasing/increasing without the equation by using the given graph.

4. Define the integral when it is concave down/concave up: The student is asked to define when the integral function is concave down/concave up without the equation by using the given graph.

Strategies
Knowledge of derivative and integral concepts are encouraged to ensure success on this exercise. 1. The fundamental theorem of calculus states: the derivative of the integral of a function is equal to the original equation.

2. When you apply the fundamental theorem of calculus, all the variables of the original function turn into x.

3. Slope intercept form is: y=mx+b

Point-slope form is: y-y1 = m(x-x1)

4. The integral is decreasing when the line is below the x-axis and the integral is increasing when the line is above the x-axis.

5. The integral is concave down when the line is decreasing and the integral is concave up when the line is increasing.

Real-life Applications
1. The fundamental theorem of calculus can be used to find the area under a continuous graph and find the tangent line at any given point of a continuous graph.