📖 Definition
What is a Tangent?
It's a line that 'kisses' a curve.
↳ A tangent shows the direction of the curve at a single point.
Kit Library / Mathematics / Calculus
⚡ Topic Learning KitDrawing Tangents to Curves
English 16 leveled MCQs 15 flashcards 9 games Free
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📖 Definition
⚙️ Process
⚠️ Common Mistake
🛠️ Application
⚙️ Process
✏️ Example
🎯 Exam Tip
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📖 Definition
⚙️ Process
Steps to Draw a Tangent
Master this simple 3-step drawing technique.
↳ Precision in ruler placement is key to an accurate tangent.
⚠️ Common Mistake
Avoid Crossing the Curve
Don't let your tangent cut through!
↳ A true tangent touches, it doesn't intersect.
🛠️ Application
Tangents Find Instantaneous Rate of Change
Unlock the 'speed at an instant'.
The primary application of drawing a tangent is to find the instantaneous rate of change of a quantity represented by a curve at a specific point.
↳ The slope of the tangent reveals how fast something is changing at that exact moment.
⚙️ Process
Calculating Gradient of a Tangent
Turn your drawn line into a number.
↳ The gradient formula quantifies the steepness of your tangent line.
✏️ Example
Finding Instantaneous Speed
How fast were you going at 5 seconds?
↳ A tangent's gradient on a distance-time graph directly gives instantaneous speed.
🎯 Exam Tip
Practice Makes Perfect
Your drawing skill matters in exams.
Drawing accurate tangents is a visual skill that improves significantly with practice. Examiners look for tangents that are clearly touching at only one point and represent the curve's direction well.
↳ Regular practice ensures precision and confidence in drawing tangents for calculations.
Smart notes
1
What you'll study, topic by topic
Drawing Tangents to Curves
15 minThis topic explains how to draw a tangent line to a curve at a specific point. It covers the definition of a tangent, the step-by-step process for drawing one, and its application in finding instantaneous rates of change…
- A tangent is a straight line that touches a curve at exactly one point, called the **point of tangency**.
- The tangent line does not cross the curve at the point of tangency; it only touches it.
- To draw a tangent, first locate the specific point on the curve.
Full explanation, worked examples & memory tricks in the app
Leveled MCQ practice
Try the smart MCQs from this kit
All 16 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Warm-up · 6 questions
Q1. Which statement best describes a tangent line to a curve?
Beginner
A
A line that touches the curve at exactly one point.
B
A line that crosses the curve at two points.
C
A line that lies entirely above the curve.
D
A line that connects the endpoints of the curve.
Answers and explanations are in the Veda app.
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?
Beginner
A
It is a normal line to the curve.
B
Nothing is wrong; tangents may cross at P.
C
It is a tangent because it passes through P.
D
It is a secant, not a tangent, at P.
Answers and explanations are in the Veda app.
Q3. On a distance-time graph, the gradient of the tangent at a point gives:
Beginner
A
Total distance travelled
B
Acceleration at that point
C
Average speed over the whole journey
D
Instantaneous speed at that point
Answers and explanations are in the Veda app.
Q4. To calculate the gradient of a tangent line, which two points should you use?
Beginner
A
Any two points on the tangent line
B
The endpoints of the curve
C
Any two points on the original curve
D
The point of tangency and the origin
Answers and explanations are in the Veda app.
Q5. Which of the following is the correct formula for the gradient of a line through $(x_1, y_1)$ and $(x_2, y_2)$?
Beginner
A
$m = (y_2 - y_1)(x_2 - x_1)$
B
$m = \frac{y_2 + y_1}{x_2 + x_1}$
C
$m = \frac{y_2 - y_1}{x_2 - x_1}$
D
$m = \frac{x_2 - x_1}{y_2 - y_1}$
Answers and explanations are in the Veda app.
Q6. A tangent line touches a curve at only one point, the point of tangency.
Beginner
A
True
B
False
Answers and explanations are in the Veda app.
Building up · 5 questions
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?
Intermediate
A
The instantaneous vertical velocity at $t = 2$ s
B
The total time of flight
C
The average velocity over the whole flight
D
The maximum height reached
Answers and explanations are in the Veda app.
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?
Intermediate
A
$0.5$
B
$3$
C
$2$
D
$6$
Answers and explanations are in the Veda app.
Q9. Why does the tangent line give the instantaneous rate of change rather than the average rate of change?
Intermediate
A
Because it crosses the curve at many points.
B
Because it uses the endpoints of the curve.
C
Because it touches the curve at only one point, capturing the behaviour at that exact point.
D
Because it is always horizontal.
Answers and explanations are in the Veda app.
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?
Intermediate
A
The instantaneous rate of change at the point of tangency
B
Zero
C
The average gradient between those two curve points, not the tangent's gradient
D
The correct tangent gradient
Answers and explanations are in the Veda app.
Q11. Which of the following best explains why the tangent line is useful for finding instantaneous speed?
Intermediate
A
It connects the start and end points of the graph.
B
Its gradient equals the speed at the exact point of tangency.
C
It gives the average speed over the whole journey.
D
It is always parallel to the x-axis.
Answers and explanations are in the Veda app.
Topper level · 5 questions
Q12. A tangent is drawn to a curve at point P. Which statement about the tangent near P is true?
Advanced
A
It must cross the curve at P.
B
It touches the curve only at P in the immediate vicinity of P.
C
It must be vertical at P.
D
It must touch the curve at two points near P.
Answers and explanations are in the Veda app.
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?
Advanced
A
Speed is the same at both times.
B
Speed cannot be compared from the curve.
C
Speed is greater at $t=3$ s than at $t=7$ s.
D
Speed is greater at $t=7$ s than at $t=3$ s.
Answers and explanations are in the Veda app.
Q14. A tangent line at point P has gradient $0$. What does this tell you about the curve at P?
Advanced
A
The curve is vertical at P.
B
The curve has a sharp corner at P.
C
The curve is undefined at P.
D
The curve is momentarily horizontal at P.
Answers and explanations are in the Veda app.
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?
Advanced
A
Both are tangents because both pass through the point
B
Student A's line, because it passes through the point
C
Neither is a tangent
D
Student B's line, because it only touches the curve at the point
Answers and explanations are in the Veda app.
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?
Advanced
A
$1.5$ units per second
B
$3$ units per second
C
$2$ units per second
D
$6$ units per second
Answers and explanations are in the Veda app.
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