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Key Calculus Terms

• Calculus: Branch of mathematics that addresses the use of limits as tools to solve problems.
  
• Differential Calculus: Branch of mathematics that uses limits to develop derivatives and differential equations.
  
• Integral Calculus: Branch of mathematics that uses limits to find integrals.
  
• Slope (of a line): Change in y values divided by change in x values for any two points on a given line.
  
• VANG: Acronym (initials) representing the four mode through which a concept should be explored and mastered in mathematics.

• How do we find the rate of change for any graph?
  
• Example:  What is the slope of this graph, or rate of change of the graph, at any point?

• Comparison of ability to solve problems with and without Calculus:

With Calculus	Without Calculus
Find f(x), the limit of f(x) and the derivative of f(x)	Find f(x)
Find the slope of the curve at any x	Find the slope of a line
Find the rate of change at any instant	Find only average rates of change
Curvature of any curve	Curvature of a circle

• How do we find the area of an irregularly shaped object?
  
• It is possible to approximate the area of a figure by using smaller figures, such as triangles and rectangles, to approximate the figure.

• We can break the figure into a number of rectangles and find the area of each:
  3x2 + 3x4 + 3x6.5 + 3x11 = 70.5 square units

• This may be done alternatively:
• 3x2 + 3x4 + 3x6.5 = 37.5 square units
• We can average this and the previous result to get a better approximation than either alone: 59 square units

Comparison of ability to solve
problems with and without Calculus:

With Calculus	Without Calculus
Area under a curve	Area of a common, regular shape
Volume of a region under a surface	Volume of a common solid
Length of an arc	Length of a straight line segment
Sum of an infinite nuber of terms	Sum of a finite number of terms

Similarities:

• Calculus is a branch of mathematics where the basic principals fit a rigid structure.  Throughout the course you will learn important, bedrock ideas like:
  
  • The Mean Value Theorem
    
  • Rolle’s Theorem
    
  • The First Fundamental Theorem of Calculus
    
  • The Second Fundamental Theorem of Calculus
    
  • Continuity
    
  • Differentiability
    
• Calculus is a branch of mathematics that builds on itself.  Like links in a chain, later concepts will build on concepts from earlier in the course. 

Differences:

• Calculus is more a system built on theorems, concepts and principles – not recipes to perform
  
• Calculus is oriented more toward problem solving and less toward basic mathematical skills

• Represent mathematical concepts verbally, analytically, numerically and graphically (VANG)
  
• You will be better at many “pre calculus” skills because of the frequency of their use – be patient and value them
  
• Calculus is best practiced with others who are at close to your ability level.  Find 2 – 5 people who are reliable and study weekly with them
  Calculus was created to solve specific problems.  You will find your problem solving skills improving, but must be persistent, especially at first.
  
• Calculus texts are difficult to read.  Focus on examples and solving practice problems!
• The fact that you are committed to working through our tutorials is a credit to your desire and dedication.  This course was designed with the same desire and dedication, along with research-based, time tested methods.  YOU CAN DO IT – WE WILL HELP!