2022 HSC Mathematics Extension 1
Question 12(c):
Tangents and gradients
Find the equation of the tangent to the curve at the point with coordinates . Give your answer in the form .
How to recognise this question
Question type: tangents gradients
- The command and notation point to tangents gradients.
- The requested response is a worked solution.
How to handle it: differentiate to find the gradient, then use the point-gradient form of the tangent.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Differentiate with the product rule: .
- At : .
- Point-slope: .
- Simplify: .
- NESA: 3 marks correct; 2 for correct slope; 1 for attempting the derivative of .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Find the tangent to at .
Question 2
For , find at and the tangent there.
Question 3
Show that the gradient of at is .