Differentiating and integrating components | Year 12

01 / LEARN

The idea, step by step

Work component by component

The derivative of $r = (x, y, z)$ is $(x^{'}, y^{'}, z^{'})$ . Apply ordinary scalar rules, including chain-rule factors, to each corresponding component.

Integrating introduces a vector constant $(C_{1}, C_{2}, C_{3})$ . Each component has an independent constant; one scalar constant cannot generally serve all coordinates.

Check a recovered position

Integrate velocity to obtain a position family, then substitute the initial time to determine each constant. Differentiate the result and check the initial vector separately.

Definite vector integration gives displacement: $\int_{a}^{b} v (t) d t = r (b) - r (a)$ . It does not give total distance along a curved path.

A common trap

The integration constant is identical in all coordinates.

Instead: Each coordinate has its own constant.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Differentiate $(t^{3}, e^{2 t}, sin 3 t)$ .

Use the power rule on the first coordinate.
The exponential and sine coordinates need chain factors 2 and 3.
Retain component order.

(3 t^{2}, 2 e^{2 t}, 3 cos 3 t)

Example 2

Worked example

Given $v = (2 t, 3, 4 t^{3})$ and $r (0) = (1, - 2, 5)$ , find position.

Integrate to $(t^{2} + C_{1}, 3 t + C_{2}, t^{4} + C_{3})$ .
At zero the constants form (1, -2, 5).
Differentiate to check the velocity.

(t^{2} + 1, 3 t - 2, t^{4} + 5)

Check your understanding

Differentiate $(cos 2 t, t^{2})$ .

Show answer

$(- 2 sin 2 t, 2 t)$

Apply the chain rule to the cosine coordinate.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Differentiate $(t^{4}, ln t)$ for t>0.
2 marks
Find the second derivative of $(t^{3}, 2 t^{2}, cos t)$ .
2 marks
Integrate $(3 t^{2}, 2 t, 1)$ with initial position (0, 1, -1) at zero.
2 marks
Evaluate $\int_{0}^{2} (t, 2 t, 3) d t$ .
2 marks
Given $v = (e^{t}, cos t)$ and $r (0) = (2, 0)$ , find position.
2 marks
Evaluate $\int_{0}^{π / 2} (cos t, sin t) d t$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate $(t^{4}, ln t)$ for t>0.
2 marks
Use the power rule.
The logarithmic coordinate requires a positive domain.
$(4 t^{3}, 1 / t)$
Find the second derivative of $(t^{3}, 2 t^{2}, cos t)$ .
2 marks
The first derivative is $(3 t^{2}, 4 t, - sin t)$ .
Differentiate again.
$(6 t, 4, - cos t)$
Integrate $(3 t^{2}, 2 t, 1)$ with initial position (0, 1, -1) at zero.
2 marks
Integrate each coordinate.
Use one constant for each initial coordinate.
$(t^{3}, t^{2} + 1, t - 1)$
Evaluate $\int_{0}^{2} (t, 2 t, 3) d t$ .
2 marks
Antiderivative $(t^{2} / 2, t^{2}, 3 t)$ .
Evaluate upper minus lower.
$(2, 4, 6)$
Given $v = (e^{t}, cos t)$ and $r (0) = (2, 0)$ , find position.
2 marks
Integrate to $(e^{t} + C_{1}, sin t + C_{2})$ .
Initial values give $C_{1} = 1, C_{2} = 0$ .
$(e^{t} + 1, sin t)$
Evaluate $\int_{0}^{π / 2} (cos t, sin t) d t$ .
2 marks
Use antiderivative $(sin t, - cos t)$ .
Subtract values at both endpoints.
$(1, 1)$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics topic 6.3–6.5: polynomial and trigonometric parameterisations, related coordinate rates, motion, tangent directions and arc length. This is school Vector calculus, not vector-field calculus.

Official curriculum source ↗

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