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Key ideas
Key ideas
Examining rates of change close to turning points helps to identify intervals where the function increases/decreases, and identify the concavity of the function.
Numerical integration can be used to approximate areas in the physical world.
Mathematical modelling can provide effective solutions to real-life problems in optimization by maximizing or minimizing a quantity, such as cost or profit.
I can explain the relationship between a function, its derivative and the second derivative in terms of maxima, minima, points of inflexion and roots.
Finding local maxima and minima
Finding local maxima and minima
Inflexions and Shape
Inflexions and Shape
Concavity and Inflexions
http://youtube.com/watch?v=c1N8zyVhWxM
Concavity and Second Derivatives - Examples of using the second derivative to determine where a function is concave up or concave down.
Optimisation
Optimisation
Some examples
http://youtube.com/watch?v=EOJbmMB8uCQ
An Optimization is Shown using Derivatives
http://youtube.com/watch?v=Zq7g1nc2MJ8
Another Optimization Problem using Derivatives!
We use the second derivative test to classify stationary points, that is : to determine their nature. We start by stating the rule and then work through an example in which we find a function's stationary points and use the second derivative test to classify them. Given a function has a stationary point when x = c then: - if f''(c) is negative then the stationary point is a maximum - if f''(c) is positive then the stationary point is a minimum - if f''(c) equals zero then the test is inconclusive and we should study the sign of the first derivative to classify the stationary point.
We see how to study the sign of the derivative to classify, or determine the nature of, stationary points: maximum, minimum, or horizontal points of inflexion, without a calculator, with a sign table. This method is sometimes called the first derivative test. This is the first of several tutorials on the first derivative test.
https://www.youtube.com/watch?v=YjdsFu2smUA
We use the sign of the derivative, to classify a function’s stationary points: maximum, minimum or horizontal point of inflexion? We use a sign table and do this by hand (without a calculator). The method is known as the first derivative test.
https://www.youtube.com/watch?v=FKfbWFg_q_g
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