Integrating Twice
The first integration gives the gradient
The second derivative describes how the gradient changes. Starting from it, integrate once to obtain , then again to obtain .
Each integration introduces a constant. Call them and so that they are not confused. The first constant becomes at the next integration, since .
Use a gradient condition and a height condition
Worked example
Find f(x) if f″(x) = 12x and (1, 5) is a stationary point.
Stationary means . Use that before the height .
Answer
Differentiating twice returns . The first derivative is zero at 1 and the height there is 5: all conditions have been checked.
Common mistake
Replacing both constants by one loses information. A point alone cannot generally determine both constants; a stationary point supplies both a gradient and a height.
Your turnPractice
A curve has and tangent at . Find the curve.
Show worked answer
The tangent gives gradient −1 and point .
Answer