Achievement Standard 2.3 Calculus

Differentiation investigates the slope of graphs, especially curves. Tangent and normal lines to a curve can be found. The turning points on graphs (which have zero gradient) can be found and the maximum and minimum values. It also involves the rates of change of variables - such as finding the speed from a distance graph which is a curve.

Integration investigates the areas under curves between set boundaries. It also involves rates of change.

Achievement Objectives Topic overview: Gradient of a curve at a point, tangents, turning points, areas under curves.

Review of the gradient for straight lines. The gradient for curves

Calculus Surfer | www.univie.ac.at Differentiation puzzles matching the graph and the derivitive function

Demo of Gradient Function (enable macros)

Introduction to differentiation and motion graphs link to www.mathsroom.co.uk

Gradient of curves

Gradient of a line segment

How can we find an equation to calculate the gradient of a curve at any given point? Differentiate.

Differentiation using limit as h --> 0 then simple differentiation.

Demonstration of differentiation & graphs (Contains macros)

Gradient of a parabola
Gradient of a cubic

Differentiation practice and finding the derivitive of fractional & negative powers

Powerpoint: Basic Differentiation

Differentiation

Differentiating and finding the gradient at a point (given an 'x' value)

Using the derivitive and a given 'x' value or a given point, determine the equation of the tangent line.

Identify features of graphs: where the graph is increasing, decreasing, points of inflection and stationary points.

Use calculus to find local maximum, local minimum, and points of inflection. Turning points

Using differentiation techniques to determine maximum valuse, optimal soutions, of minmum values. Max & Min applications

Skills: Extract relevant information from a word problem, form an equation, differentiate and solve the problem.

The area under a line graph can be easily found. A curve is more challanging. Approximating the area using the trapezium rule.

Trapezium rule

Trapezium

Antidifferentiation: introducing the notation involved and the process of antidifferentiation, relating this to the graph of a function. Practicing skills

Finding the original equation when given a derivitive and point on the original function. Practice finding C

Calculating a definite integral using the anti-derivitive and the limits of integration. Calculating the area between a function and the x-axis between two 'x' values.

Definite Integrals

When the graph of a function crosses the x or y-axis more care is needed. the integral should be split into parts, then combined. The area below the axis is calculated to be a negative area so must be changed to a positive value.

The area bound between two functions can be calculated by generating a new function (the difference between the two origal functions) and integrating between the two limits (or points of intersection between the functions.

Application problems (excellence) often involve:
1) Forming a suitable function to model a practical situation,
2) Determining the limits for intergration - often involving solving simultaneous equations,
3) Calculating the definite integral,
4) Answering the question - such as calculating a volume when integration was used to obtain a cross-sectional area

Review graphing skills of forming a function (often a parabola) to fit a situation (vertical & horizontal movement & vertical stretch)

The are many formulas which can be differentiated to form a rate of change function. these are often measurement formula.

Linking distance, speed, and acceleration functions using differentiation and anti-differentiation

Kinematics

Word problems and applications of differentiation and integration. Form a function, decide if differentiation or integration is appropriate, calculate and solve the problem.