The antiderivative · Grade 10 · 5 units
In The antiderivative (Grade 10 · 5 units) you practise 4 skills: Finding an antiderivative · Many antiderivatives · An antiderivative through a point · Reading the antiderivative from a graph. An antiderivative of a function is a function whose derivative is the given one. To find it, undo the rules of differentiation: in each term raise the power by one and divide by the new power, and a number on its own becomes that number times x. For example, an antiderivative of two x is x squared.
A function has many antiderivatives. The derivative of a constant is 0, so any constant can be added to an antiderivative. Any two antiderivatives of the same function differ by a constant, because the derivative of their difference is 0. One point on the graph fixes the constant. Where the given function is positive the antiderivative increases, and where it is negative the antiderivative decreases. Why do the rules hold? The derivative of a sum is the sum of the derivatives, because the average gradient of a sum is the sum of the average gradients, and so is the limit. The derivative of a number times a power of x is the number times the power, times x to the power one less. Both rules follow from the definition of the derivative as a limit. Finding an antiderivative undoes differentiation: differentiate what you found and you are back at the given function. A model proof, from the definition of the derivative, that the derivative of x squared is two x: the average gradient between x and x plus h is the difference of the squares over h. That difference is two x times h plus h squared, so the average gradient is two x plus h. As h tends to 0, the average gradient tends to two x. So x squared is an antiderivative of two x.
Tip: At the end, differentiate what you found: if you get the given function back, the answer is right.
What you learn
Finding an antiderivative
For example: Find an antiderivative of f(x) = 8x .
Answer and solution
The answer: 4x²
In each term raise the power by one and divide by the new power: F(x) = 4x² . Check: F′(x) = 8x . Adding any constant gives another antiderivative.
Many antiderivatives
For example: The function F(x) = −3x² is an antiderivative of f(x) = −6x . Which of these is also an antiderivative of it?
- −3x² + 9x
- −6x
- −6
- −3x² + 9
Answer and solution
The answer: −3x² + 9
Differentiate each option. The derivative of −3x² + 9 is −6x , because the derivative of a constant is 0. So the answer is −3x² + 9 .
An antiderivative through a point
For example: Here F′(x) = 6x and F(1) = −4 . Find F(3) .
Answer and solution
The answer: 20
Every antiderivative is F(x) = 3x² + C . Put in the point: 3 × 1² + C = −4 , so C = −7 . Now F(3) = 3 × 3² − 7 = 20 .
Reading the antiderivative from a graph
For example: The function F is an antiderivative of f(x) = (x + 4)(x − 4) . What does F do for −4 < x < 4 ?
- It increases
- It stays the same
- It decreases
Answer and solution
The answer: It decreases
The derivative of F is f . The interval has no root, so one point is enough: f(−3) = −7 . The value is negative, so the antiderivative decreases throughout.
Not on the site yet
Parts of this topic that are not built yet: The definite integral and area · Proving the rules of differentiation.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).