Intervals of Increase and Decrease
Finding intervals of increase and decrease of a function can be done using either a graph of the function or its derivative. These intervals of increase and decrease are important in finding critical points, and are also a key part of defining relative maxima and minima and inflection points.
Maxima and Minima
Finding relative maxima and minima of a function can be done by looking at a graph of the function. A relative maximum is a point that is higher than the points directly beside it on both sides, and a relative minimum is a point that is lower than the points directly beside it on both sides. Relative maxima and minima are important points in curve sketching, and they can be found by either the first or the second derivative test.
Inflection Points
Inflection points are points on the graph where the concavity changes. A positive second derivative means a function is concave up, and a negative second derivative means the function is concave down. These inflection points are places where the second derivative is zero, and the function changes from concave up to concave down or vice versa.
Curve Sketching
Sketching a curve from knowledge of the signs of the first and second derivatives is a useful way to find the approximate shape of a function's graph. When curve sketching making a sign chart of the derivatives is an easy way to spot possible inflection points and to find relative maxima and minima, which are both key in sketching the path of a function.
Using the First Derivative Test
Some optimization problems use the first derivative test to find an absolute minimum or maximum. Using the first derivative test requires the derivative of the function to be always negative on one side of a point, zero at the point, and always positive on the other side. Other methods of solving optimization problems include using the closed interval method or the second derivative test.
Application : Cost and Revenue
One of the most important parts of economics is knowing the revenues and costs and how they relate to increased production. These can both be modeled by functions. These cost and revenue functions can then be manipulated like any other function. The profit is the difference between total revenue and total cost.
Application :Calculus Optimization Problems
Some economics problems can be modeled and solved as calculus optimization problems. These problems usually include optimizing to either maximize revenue, minimize costs, or maximize profits. Solving these calculus optimization problems almost always requires finding the marginal cost and/or the marginal revenue.
Critical Points
Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero. All relative maxima and relative minima are critical points, but the reverse is not true.
First Derivative Test
The first derivative test is a way to find if a critical point of a continuous function is a relative minimum or maximum. Simply, if the first derivative is negative to the left of the critical point, and positive to the right of it, it is a relative minimum. If the first derivative test finds the first derivative is positive to the left of the critical point, and negative to the right of it, the critical point is a relative maximum.
Second Derivative Test
The second derivative test is useful when trying to find a relative maximum or minimum if a function has a first derivative that is zero at a certain point. Since the first derivative test fails at this point, the point is an inflection point. The second derivative test relies on the sign of the second derivative at that point. If it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum.
Closed Interval Method
The closed interval method is a way to solve a problem within a specific interval of a function. The solutions found by the closed interval method will be at the absolute maximum or minimum points on the interval, which can either be at the endpoints or at critical points. Other ways of solving optimization problems include using the first derivative test or the second derivative test.
Using the Second Derivative Test
Some optimization problems can be solved by use of the second derivative test. If the second derivative is always positive, the function will have a relative minimum somewhere. If it is always negative, the function will have a relative maximum somewhere. Other ways of solving optimization problems include using the closed interval method or the first derivative test.
Application : Marginal Cost and Revenue
The marginal cost and marginal revenue are the additional amount of cost or revenue that arise from producing one more item. If you take the derivative of the cost and revenue functions, you get approximately the marginal cost and revenue. Marginal cost and revenue are useful in solving calculus optimization problems involving economics.
