Mathematica Labs | Denis Shubleka

Subject: Calculus

Topic: Discovering FTC

Goal: Use Mathematica to introduce the Fundamental Theorem of Calculus.

Task 1

a) Draw or plot the line y = 2t + 1, and use geometry to find the area under the 
line, above the t-axis, and between the vertical lines t=1 and t=3.

Plot[2t+1,{t,1,3},PlotRange->{-1,8},Filling->Axis]

b) If x > 1, let A(x) be the area of the region that lies under the line y = 2t + 1,
between t=1 and t=x. Sketch this region on paper, and use geometry to find an 
expression for A(x). The Manipulate command below varies x from 1.1 to 3.

Manipulate[Plot[2t+1,{t,1,x},PlotRange->{-1,8},Filling->Axis],{x,1.1,3}]

c) Differentiate the area function A(x) with respect to x. What do you notice? If using Mathematica, execute the following commands in the given order:

A[x_]:=(((3+(2x+1))*(x-1))/2);
D[(((3+(2x+1))*(x-1))/2),x]

Simplify[%]

Task 2

a) If x ⩾ -1, define A(x)=(∫_(-1)^x)(1+(t^2))dt. A(x) represents the area of a region. Sketch
that region on paper.

b) Use Mathematica to find an expression for A(x)

Integrate[1+(t^2),{t,-1,x}]

c) Differentiate the answer using:

D[%,x]

What do you notice?

d) If x ⩾ -1 and h is a small positive number, then A(x+h) - A(x) represents the area
of a region. Describe and sketch the region on paper.

e) Draw a rectangle that approximates the region above. By comparing the areas of these
two regions, argue that:

((A(x+h)-A(x))/h)≈1+(x^2)  
f) Give an intuitive explanation for the result of part c). [Hint: Consider the limit
of the left hand side in the part e), as h approaches 0.]

Task 3

a) Draw the graph of the function f(x)=cos((x^2))in the viewing rectangle [0, 2] by [-1.25, 1.25].

f[t_]:=Cos[(t^2)]

Plot[f[x],{x,0,2},PlotRange->{-1.25,1.25}]

c)Define a new function, g(x), using f(x) as the integrand:

g[x_]:=(∫_0^x)f[t]\[DifferentialD]t

Test its functionality:

g[1]//N

d)Plot g’(x) and f(x) in the same window:

Plot[{g'[x],f[x]},{x,0,2}]

What do you notice? Ask Mathematica to onfirm your observation:

g'[x]==f[x]

e) Now it is time to generalize. Suppose that f is a continuous function on
a closed interval [a, b]. If we define a new function:
g(x)=(∫_(t=a)^(t=x))f(t)dt,  conjecture an expression for g’(x).

The result is one of the two parts of the Fundamental Theorem of Calculus.

Related Exercises/Notes:

ap-calc.github.io