Drawing Tangents to Curves: 16 MCQs

Q1. Which statement best describes a tangent line to a curve?

Q2. A student draws a line that touches a curve at point P but also cuts through the curve at P. What is wrong with this line?

Q3. On a distance-time graph, the gradient of the tangent at a point gives:

Q4. To calculate the gradient of a tangent line, which two points should you use?

Q5. Which of the following is the correct formula for the gradient of a line through $(x_1, y_1)$ and $(x_2, y_2)$?

Q6. A tangent line touches a curve at only one point, the point of tangency.

Q7. A ball is thrown upward and its height-time graph is plotted. What does the gradient of the tangent at $t = 2$ s represent?

Q8. A tangent is drawn at point P on a curve. Two points on the tangent line are $(1, 3)$ and $(4, 9)$. What is the gradient of the tangent?

Q9. Why does the tangent line give the instantaneous rate of change rather than the average rate of change?

Q10. A student calculates the gradient of a tangent using two points that lie on the curve but not on the tangent line. What is the result?

Q11. Which of the following best explains why the tangent line is useful for finding instantaneous speed?

Q12. A tangent is drawn to a curve at point P. Which statement about the tangent near P is true?

Q13. A distance-time curve is steep at $t=3$ s and flat at $t=7$ s. What can be said about the instantaneous speed at these times?

Q14. A tangent line at point P has gradient $0$. What does this tell you about the curve at P?

Q15. Two students draw tangents at the same point on a curve. Student A's line crosses the curve at that point; Student B's line only touches it. Which line is the correct tangent?

Q16. A tangent is drawn at a point on a height-time graph. The tangent passes through $(2, 5)$ and $(4, 11)$. What is the instantaneous vertical velocity at that point?