An Introduction to Calculus

1
An Introduction to Calculus
2
Calculus

• Study of how things change
• Allows us to model real-life situations very
  accurately

3
Topics (not in order)

• Limits
• Slope of a curve
• Tangent Lines
• Extreme values of a curve
• Area under a curve/volume of a region
• Lengths
• Riemann Sums (Finite Infinite)
• Rates of Change

4
Before Calculus After Calculus

• Find slope of a line
• Calculate average speed of a car
• Calculate area/volume of rigid geometric figures

• Find slope of a curve at a point
• Determine exact speed of a car at a particular
  time
• Determine area/volume of any figure under any
  curve/region

5
Two Main Problems to Solve

• I. Tangent Line Problem
• How do you find the equation of the tangent
  line to the graph of a curve at a particular
  point?
• II. Area Under Curve Problem
• How can you find the area underneath a curve and
  above the x-axis?

6
Secant line
f(cDx)
Secant line
f(cDx)
Tangent line
f(c)
c
cDx
cDx
7
As Dx decreases (Dx?0), secant line approaches
the tangent line.
8
Dx
Dx
Dx
Dx
a
b
x1
x2
x3
9
Dx
Dx
Dx
Dx
Dx
Dx
Dx
Dx
a
b
x1
x2
x3
x4
x5
x6
x7
10
As the number of rectangles increases, the sum of
their areas approaches the exact area of the
curve.