- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Differentiation rules practice question 13
The curve has a tangent line parallel to at exactly one point with . Find the -intercept of that tangent line.
What do parallel lines have in common? Use that to find where on the curve the tangent touches, before you think about the intercept.
Parallel lines have equal slopes, so set and solve for .
Keep the solution with x > 0, find the point on the curve there, and write the tangent line through it.
A number.
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Worked solution
Strategy
A tangent parallel to y = 9x + 1 has slope 9. Solve y′ = 9 for the point of tangency, then build the tangent line and read off its intercept.
Working
Equal slopes; divide by 3 and factor.
Only x = 3 is positive, so the point of tangency is (3, 5).
Point-slope form; the intercept is the constant term.
Answer
The answer is the tangent line is y = 9x − 22, with y-intercept −22.
Check
The curve minus the line is x³ − 3x² − 9x + 27 = (x − 3)²(x + 3), with a double root at x = 3, so the line touches there.
Where this goes wrong
Stopping at x = 3, or giving the height y(3) = 5. The question asks where the tangent line crosses the y-axis, which needs the whole line.
Try a similar question
Find an equation of the tangent line to at the point where .Saved in this browser only.