Tangent equation and greatest and least value on an interval
Write the tangent equation with y = f(x₀) + f′(x₀)(x − x₀), find a tangent parallel to a given line and one through a point off the curve, then find the greatest and least value of a function on [a;b].
- Germany: grade 11
- Ukraine: grade 10
- Russia: grade 11
Guide
In the Ukrainian programme (Grade 10, algebra and analysis, § 3, items 21 and 24) the tangent to the graph of f at the point with abscissa x₀ is y = f(x₀) + f′(x₀)(x − x₀). Its slope is f′(x₀).
For a tangent parallel to a line, set f′(x₀) equal to the slope of that line. For a tangent through a point that is not on the graph, call the point of contact x₀ and substitute the given point into the tangent equation.
Greatest and least value of a continuous function on [a;b]: find the critical points inside the interval, compute f there and at the ends a and b, then pick the largest and the smallest.