The limit of f(x) = x² as x tends to 3 is 9 and dy/dx for y = x³ is 3x², so this IB sheet opens by taking a limit and then a derivative. The key records 6 as a gradient answer and 0 as a derivative result.
Key Takeaways
The limit of x² as x tends to 3 is 9 (Q1).
dy/dx = 3x² when y = x³ (Q2).
Q4 records 6 as a gradient value.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.What is the limit of f(x) = x² as x tends to 3?
(a) 6(b) 9(c) 3(d) 27
2.What is dy/dx if y = x³?
(a) 3x²(b) x²(c) 3x(d) 2x³
3.What is the gradient of the tangent to y = x² at x = 2?
(a) 2(b) 4(c) 8(d) 1
4.If distance is s(t) = t², what is the velocity at t = 3?
(a) 6(b) 3(c) 9(d) 2
5.What is the limit of (x² − 1) ÷ (x − 1) as x tends to 1?
(a) 1(b) 0(c) 2(d) ∞
6.What is dy/dx if y = 5?
(a) 5(b) 0(c) 5x(d) 1
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Find dy/dx if y = 4x³.
8.Find the gradient of y = x² − 3x at x = 1.
9.Evaluate the limit of (x² − 4) ÷ (x − 2) as x tends to 2.
10.If f(x) = √x, find f'(4).
11.Find dy/dx if y = 7x² − 4x + 1.
12.A car's distance is s(t) = 5t² metres. Find its velocity at t = 2 seconds.
13.Find the equation of the tangent to y = x² at x = 1.
14.A stone's height is h(t) = 20 − 5t² metres. Find its velocity at t = 1 second.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A ball's height is h(t) = 20t − 5t² metres. Find the maximum height reached.
16.A firm's cost is C(x) = 100 + 5x + 0.1x² dollars. Find the marginal cost at x = 10.
17.A population is given by P(t) = 1000 + 50t + 2t². Find the rate of growth at t = 5.
18.A sphere has volume V(r) = (4 ÷ 3)πr³. Find dV/dr at r = 3.
19.A particle's velocity is v(t) = t² − 4t + 3. Find the times when the velocity is zero.
20.A room's temperature is T(t) = 20 + 10t − t² degrees after t hours. Find the rate of change at t = 3.
Answer Key
1. b) 9
2. a) 3x²
3. b) 4
4. a) 6
5. c) 2
6. b) 0
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
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14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the limit of x² as x tends to 3?
9, option (b) in this worksheet (Q1).
What is dy/dx when y = x³?
3x², option (a) in this worksheet (Q2).
What is the derivative of a constant?
0, option (b) in this worksheet (Q6).
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