Curve sketching and optimisation
Find the asymptotes and key points of a graph, then use a maximum to solve problems such as the largest area and the largest volume.
- UK: Year 12
- Germany: grade 11
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 11
- Palestine: grade 12
- Ukraine: grade 10
- Russia: grade 11
Guide
To sketch a graph, ask: where is the denominator zero (a vertical asymptote)? What does f approach when x is very large (a horizontal asymptote)? Where is the derivative zero (peaks and valleys)? Where does f″ change sign (a point of inflection)?
In an optimisation problem, first write the quantity you want to make largest or smallest in terms of one variable. Then put its derivative equal to zero, check the solution and work out the value of the function there.
The answer is usually the value of the function (the largest area), not x. Remember to put x back into the function.