Drawing Tangents to Curves: 20 flashcards
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1. What is a tangent line?
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A straight line that touches a curve at exactly one point without crossing it at that point.
2. What does the gradient of a tangent line represent?
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The instantaneous rate of change of the curve at the point of tangency.
3. How do you draw a tangent to a curve?
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1. Locate the point of tangency. 2. Position a ruler to touch only that point. 3. Draw the line along the ruler's edge.
4. What formula is used to calculate the gradient of a tangent line?
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$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
5. On a distance-time graph, what does the gradient of the tangent give?
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The instantaneous speed at that point in time.
6. On a velocity-time graph, what does the gradient of the tangent give?
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The instantaneous acceleration at that point in time.
7. What is the difference between a tangent and a secant line?
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A tangent touches the curve at one point; a secant cuts through the curve at two points.
8. Why is the tangent line important in calculus?
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It is the foundation for the derivative, which measures instantaneous rate of change.
9. What is a common mistake when drawing a tangent?
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Drawing a line that crosses the curve at or near the point of tangency instead of just touching it.
10. In economics, what does the gradient of a tangent to a total cost curve represent?
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The marginal cost at that production level.
11. In biology, what can the gradient of a tangent to a population growth curve indicate?
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The instantaneous population growth rate at that time.
12. What two points do you use to calculate the gradient of a tangent line?
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Any two distinct points on the tangent line itself, not necessarily on the curve.
13. What does 'instantaneous rate of change' mean?
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The rate at which a quantity is changing at a specific instant in time.
14. How does the tangent line help in analyzing motion?
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It gives the instantaneous speed or acceleration at a specific moment from a graph.
15. What is the point of tangency?
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The single point where the tangent line touches the curve.
16. Why does the tangent line not cross the curve at the point of tangency?
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By definition, a tangent only touches the curve at that point and does not intersect it there.
17. What is the gradient of a horizontal tangent line?
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Zero, because there is no change in y as x changes.
18. In physics, what does the gradient of a tangent to a displacement-time graph give?
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Instantaneous velocity.
19. What is the first step in drawing a tangent line?
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Identify the exact point on the curve where the tangent should be drawn.
20. How can you check if a drawn line is a good tangent?
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Ensure it touches the curve at only one point and follows the curve's direction at that point.
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