Not in the course
Why does an antiderivative need a + C?
Every curve that runs along a slope field of y′ = f(x) is the same curve moved up or down, so undoing a derivative gives a family, and one known value picks its member.
Builds on the chain rule and logarithms and exponential and trigonometric functions.
The lesson as text
1. Undo a derivative
Which function has derivative ? Read the power rule backwards, term by term: , the green curve. Each short segment behind it has slope at its point, and the curve runs along them.
Question: Is the only function whose derivative is ?
No. Moving the graph up by adds a constant, and a constant's derivative is : . The slope field depends on alone, so all the segments in a column are parallel, and every copy of moved up or down runs along them too.
The figure: The slope field of y′ = 3x² − 12x + 9: in each column every segment has the same slope. On it, the curve F(x) = x³ − 6x² + 9x. Faint copies of it, moved up and down, run along the segments too.
2. One value picks one
The antiderivatives form a family, , one curve for each constant . Drag the green curve up or down: at every height it fits the field. To pick one, you need one value of it, such as .
Drag the curve until it passes through the point .
, so . One value fixes the constant: of the whole family, only has this derivative and passes through .
The figure: The slope field of y′ = 3x² − 12x + 9: in each column every segment has the same slope. On it, the curve F(x) = x³ − 6x² + 9x + 1.0. Faint copies of it, moved up and down, run along the segments too. The point (1, 5) is marked.
3. Velocity to position
Read the figure as motion. A cart on a track has velocity m/s, the slopes of the segments, and starts m from a post: . Its position is an antiderivative of . The green curve is , with no constant.
Question: How far from the post is the cart at ?
m. and , so . Without it is , the amber curve: the right velocity at every instant, but a cart that started at the post. And is , a speed, not a place.
The figure: The slope field of ds/dt = v(t) = 3t² − 12t + 9: in each column every segment has the same slope. On it, the curve s(t) = t³ − 6t² + 9t + 2.0. The start, s(0) = 2, is marked. An amber curve, the same with no constant, starts at 0 and reaches 4.00 at t = 4, against 6.00.
4. A change ignores C
Between two times, every curve of the family rises or falls by the same amount: each is one curve moved up. Drag the curve and watch the cart's position at and at .
Question: From to , how does the cart's position change?
Back m: , and cancels. Between the two times , so the cart backs up, though it stands still at each end. A change needs no starting value; a position does.
The figure: The slope field of ds/dt = v(t) = 3t² − 12t + 9: in each column every segment has the same slope. On it, the curve s(t) = t³ − 6t² + 9t + 2.0. Marked: s(1) = 6.00 and s(3) = 2.00.