Infer primitive curves from a gradient function lesson | Mathematical Methods (Year 11, VCE Units 1 & 2, Anti-differentiation and integrals, Gradient functions and motion)

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

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
\(\displaystyle F(-1)=-1+3+2=4\)Substitute the first stationary input
\(\displaystyle F(1)=1-3+2=0\)Substitute the second stationary input
\(\displaystyle (-1,4),\ (1,0)\)The first is a local maximum and the second a local minimum

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 shift

Treating 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 inflection

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

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 11, VCE Units 1 & 2, Mathematical Methods, chapter 14: Anti-differentiation and integrals