Antiderivatives: the family F + C
About this app
A function f has infinitely many antiderivatives F + C — vertically shifted copies with the same slope f(x₀) at every point. Top: f with a ruler; bottom: the family with parallel tangents; a point P fixes C. Checking by differentiating exposes typical mistakes such as xⁿ → n·xⁿ⁻¹. Six functions from x² to eˣ.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
A function F is an antiderivative of f if F′ = f. If F is one, so is F + C, because the derivative of a constant is 0. On an interval, two antiderivatives differ only by such a constant. Their graphs are vertically shifted copies and have the same slope f(x) at every point x. Only an extra condition, such as a point P the graph must pass through, fixes C: C = y_P − F(x_P).
What the app shows
Two cards share the x-axis. The top one shows f with a ruler at x₀ and the height f(x₀). Below lies the family F + C as faint curves, with one antiderivative highlighted. At the ruler every curve has a short tangent, and all tangents are parallel, with slope f(x₀). An arrow shows the distance C to the curve with C = 0. You can pick x², 2x + 1, cos x, 1/x² for x > 0, eˣ and 3x² − 4x.
Things to try
Shift C and watch the curve slide through the family without changing its shape. Drag the ruler: the tangents stay parallel everywhere. Drag the point P, and the highlighted curve jumps to the one that passes through P. In the “Check” panel you test candidates by differentiating, including typical mistakes such as 2x instead of x³/3. A wrong candidate cuts across the tangents of the family, and G′ visibly departs from f on top.
Frequently asked questions
- Why are there infinitely many antiderivatives?
- Because a constant disappears when you differentiate: (F + C)′ = F′ = f for every number C. Each antiderivative F therefore gives a whole family F + C whose graphs are shifted vertically against each other.
- How do you find the antiderivative through a given point?
- Plug the point P(x_P | y_P) into F(x) + C and solve for C: C = y_P − F(x_P). For f(x) = cos x and P(0 | 2) this gives C = 2, so F(x) = sin x + 2. On an interval exactly one antiderivative passes through each point.
- How do you check whether an antiderivative is correct?
- Differentiate it. If you get exactly f, it is correct. A common mistake is to differentiate f instead of integrating it: x³/3 is an antiderivative of x², while 2x is its derivative and fails the check.
- What can I adjust in the app?
- You choose one of six functions, move the constant C, the ruler x₀ and the point P, switch the family, tangents and slope field on or off, and test correct and wrong candidates in the “Check” panel.
Subject: Analysis (one-dimensional)
More from Analysis (one-dimensional)
- Derivative as a sequence of secants
- Bounded growth
- Units of an integral: rate times time
- Exponential growth and decay
- Optimisation: the largest rectangle
- Area between curves
- Area function is an antiderivative
- Sequential continuity
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