Level 3 Differentiation Walkthrough
2025 NCEA Level 3 Differentiation Question 1(d)
2025 Paper
Question
The graph below shows the function \(f(x)=x^2+e^{2x}\) and the tangent to the curve when \(x=1\).
Find the point \(P\), the \(x\)-intercept of this tangent.
You must use calculus and show any derivatives that you need to find when solving this problem.
The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.
First walkthrough idea
Hint to try first
Start by finding the point on the curve when \(x=1\).
Step 1
Find the point of tangency
Substituting \(x=1\) gives \(1^2+e^2=1+e^2\).
Show the first step’s working
Substituting \(x=1\) gives \(1^2+e^2=1+e^2\).
Key result
\[ 1+e^2 \]Walkthrough overview
What this question practises
This 2025 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Finding a tangent and its x-intercept.
This is Question 1(d) from the 2025 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.
Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise finding a tangent and its x-intercept. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Use the derivative for the gradient and the original curve for the point before forming the tangent equation.