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Steve Arnold

Introducing the Differential Calculus from First Principles
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My trip to school each day takes me 20 minutes to travel 15 miles. We would say that my average speed for this trip is 45 miles per hour.
In reality, however, I speed up and slow down continuously over that time. The differential calculus allows us to describe instantaneous rates of change rather than just averages!
In this activity, we begin with the concept of gradient between two points, which lie on the graph of a function. We learn that for many functions (like my trip to school) the gradient (or rate of change) actually changes constantly and we can calculate this rate at every point of my journey, not just at start and finish.
This activity explores first principles numerically, graphically and algebraically. It has a CAS extension which uses dynamic algebra and programming.

8. The Tangent to a Curve and the Derivative of a Function
10. Geometrical Applications of Differentiation
14. Applications of Calculus to the Physical World



Steve Arnold

Exploring Newton's Method
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Suppose we wish to approximate the zeros for a function, f(x).
Newton's Method (sometimes called the NewtonRaphson Method) is an iterative method for doing this  which means that the more times we apply the method, the better our approximation should become!
This activity explores Newton's method numerically, graphically and algebraically. It has a CAS extension which uses dynamic algebra and programming.

8. The Tangent to a Curve and the Derivative of a Function
10. Geometrical Applications of Differentiation
14. Applications of Calculus to the Physical World



Steve Arnold

Introducing the Differential Calculus: The Product Rule
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The rules for differentiation for most simple functions are clear and easily learned. More difficult functions, however, can cause problems. Sometimes it helps to break the problem into smaller pieces.
This activity introduces and consolidates student skills and understanding of the Product Rule using a variety of tools, from algebraic spreadsheets to programs.

10. Geometrical Applications of Differentiation
14. Applications of Calculus to the Physical World



Steve Arnold

Introducing the Differential Calculus: The Quotient Rule
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In this activity, we consider the case of a function which can be expressed in the form u(x)/v(x).
Once again, this activity uses CAS to introduce and consolidate student skills and understanding of the Quotient Rule using a variety of tools, from computer algebra to algebraic spreadsheets to programming.

10. Geometrical Applications of Differentiation
14. Applications of Calculus to the Physical World



Steve Arnold

Introducing the Differential Calculus: Composite Functions
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In this activity, we consider the case of a function which can be expressed in the form y = f(g(x)).
Once again, this activity uses CAS to introduce and consolidate student skills and understanding of the Chain Rule using a variety of tools, from computer algebra to algebraic spreadsheets.

10. Geometrical Applications of Differentiation
14. Applications of Calculus to the Physical World



Steve Arnold

Introducing the Integral Calculus: Integration by Parts
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We have discovered that there are rules such as the Product and Quotient Rules for dealing with more difficult derivatives. But what about harder integrals?
Using CAS, algebraic spreadsheets and programming, this activity develops the method of Integration by Parts from the Product Rule, and then provides opportunities for students to explore and consolidate this important technique for integration.

11. Integration
11 Extension. Methods of Integration



Steve Arnold

Introducing the Integral Calculus: Integration by Substitution
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In this activity we examine more difficult integrals that may be approached by using appropriate substitution  but how do we recognise these integrals, and how do we tell what to substitute?
Using CAS and algebraic spreadsheets, this activity develops the method of Integration by Substitution, and then provides opportunities for students to explore and consolidate this important technique for integration.

11. Integration
11 Extension. Methods of Integration



Steve Arnold

Applications of Calculus: Simple Harmonic Motion
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In this activity we investigate an important application of calculus to the physical world, simple harmonic motion. Within the context of motion on a swing, students derive the formulas, first by differentiation of the displacement/time equation and, later, by integration from the defining equation for acceleration.
Using multiple representations and CAS, this activity provides an introduction to later work on differential equations.

14. Simple Harmonic Motion



Steve Arnold

Applications of Calculus: Torricelli's law
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Torricelli's Law describes the velocity at which water leaves a container through a (small) opening: in fact, he maintained (in 1643) that the velocity attained by the water was the same as that of water falling from the same height under the influence of gravity.
This activity uses multiple representations, algebraic programming and CAS to lead students through the derivation of Torricelli's Law and of projectile motion as applications of differential equations.

14. Projectile Motion



Steve Arnold

Applications of Calculus: Projectile Motion
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Projectile Motion can be one of the best of the applications of Calculus to explore, and in this activity we not only throw balls, we shoot cannons and even shoot for basketball hoops  and learn about differential equations at the same time.
This activity uses multiple representations, algebraic programming and CAS spreadsheets to lead students through the derivation of the defining equations for projectile motion and give plenty of opportunities for practice and consolidation of both knowledge and skills.

14. Projectile Motion



Steve Arnold

Applications of Calculus: Introducing Maclaurin Series
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Polynomials are great, aren't they? So much easier to work with than many of those other tricky functions  especially when we are doing Calculus.
This activity uses multiple representations, interactive graphs and CAS spreadsheets to lead students through the derivation of the Maclaurin and Taylor series for approximating functions, and provides plenty of opportunities for practice and consolidation of both knowledge and skills.

16. Polynomials

