From gradient to primitive
Gradient signs and zeros determine shape features; a boundary value determines the height
\[F^{\prime}(x)=g(x),\qquad F(x)=P(x)+C\]
- Transfer gradient signs A positive gradient means the primitive is increasing; a negative gradient means it is decreasing. A gradient height is a slope, not a primitive height
- Classify each zero Positive to negative gives a local maximum; negative to positive gives a local minimum. A repeated zero of a quadratic gradient gives a cubic primitive with stationary inflection
- Keep the vertical shift unknown Adding a constant changes primitive heights but preserves the gradient, stationary inputs and their classifications
- Use a boundary value Find P+C by reversing differentiation, then substitute one function value to determine C. Check both the derivative and the boundary value
Worked example
A function has derivative 3x² − 3 and passes through (0, 2). Determine its stationary points and their nature
\[F^{\prime}(x)=3(x-1)(x+1),\qquad F(0)=2\]
- \(\displaystyle F(x)=x^3-3x+C\)Reverse the polynomial derivative
- \(\displaystyle C=2\)Use F(0) = 2.
- \(\displaystyle x=-1,\ 1\)The gradient zeros are stationary inputs. Its signs are positive, negative, positive
- \(\displaystyle (-1,4),\ (1,0)\)The first is a local maximum and the second a local minimum
Common mistakes
Copying gradient heights
✕At a gradient zero the primitive height is zero✓At a gradient zero the primitive slope is zero
A function value is needed to determine the vertical shiftTreating every stationary point as a turn
✕Every zero of g is a maximum or minimum✓Inspect the gradient signs on both sides
A double zero of a quadratic gradient gives stationary inflectionDifficulty levels
introConstant or linear gradients
standardQuadratic rules and gradient graphs with simple zeros
challengeRepeated zeros and unknown vertical translation
examBoundary-selected primitive graphs, formulas and stationary coordinates
Questions about this topic
Why are sign intervals split at a repeated zero?+
The table records strictly positive or negative gradient on open intervals. F may still be increasing across the isolated stationary point as a whole
Does this topic start with a primitive and ask for its derivative?+
No. This placement starts with the gradient; forward differentiation is practised in the earlier calculus topics
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 7: Antiderivatives
In this chapter
7A Antiderivatives and arbitrary constants: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Recognise equivalent polynomial formsSimplify and combine algebraic fractionsAnti-differentiate a power of a linear function7C Integrating powers of linear functions: Recognise equivalent polynomial forms7C Integrating powers of linear functions: Solve a linear equationAnti-differentiate exponential and circular functionsAnti-differentiate by recognition from a supplied derivativeDetermine a polynomial from its derivative and a boundary conditionInfer primitive curves from a gradient functionReconstruct quantities from their ratesEvaluate and determine function rules7E Antiderivative families and applications: Solve a linear equationRound to decimal places and significant figuresScale a proportional quantity with the unitary method