When a function's derivative flips from positive to negative at x = c the curve turns a local maximum, and the tangent slope to y = x² at (1, 1) is 2, so this Class 12 sheet is all about reading derivatives as motion. The key also lists a local minimum and the answer 2πr for a perimeter problem.
Key Takeaways
A sign change from positive to negative in the derivative at x = c gives a local maximum (Q1).
The slope of the tangent to y = x² at (1, 1) is 2 (Q2).
The key records a local minimum answer.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.If the derivative of a function f changes from positive to negative at x = c, then f has a
(a) Local maximum(b) Local minimum(c) Point of inflection(d) Constant value
2.The slope of the tangent to the curve y = x² at the point (1, 1) is
(a) 1(b) 2(c) 3(d) 0
3.The rate of change of y = 3x with respect to x is
(a) 1(b) 3(c) 0(d) x
4.If f''(c) > 0 at a critical point c of a function f, then f has a
(a) Local maximum(b) Local minimum(c) Point of inflection(d) No extremum
5.The maximum value of the function f(x) = -x² is
(a) 0(b) 1(c) -1(d) 2
6.The derivative of the area A = πr² of a circle with respect to r is
(a) 2πr(b) πr(c) 2π(d) πr²
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Write the slope of the tangent to the curve y = f(x) at x = a.
8.Find the slope of the tangent to the curve y = x² at the point (2, 4).
9.Find dy/dx if y = x³.
10.Write the condition for a function f to have a local maximum at x = c.
11.Find the rate of change of y = 5x with respect to x.
12.Write the second derivative of y = x³.
13.Find the critical point of the function f(x) = x².
14.State the condition on f''(c) for a local minimum of f at x = c.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Find the equation of the tangent to the curve y = x² at the point (1, 1).
16.The radius of a circle is increasing at the rate of 2 cm/s. Find the rate of change of its area when the radius is 5 cm.
17.Find the maximum value of the function f(x) = 4x - x².
18.The edge of a cube is increasing at the rate of 3 cm/s. Find the rate of change of its volume when the edge is 10 cm.
19.Find the local minimum value of the function f(x) = x² - 4x + 5.
20.A stone thrown upward has height h = 20t - 5t² metres after t seconds. Find the maximum height it reaches.
Answer Key
1. a) Local maximum
2. b) 2
3. b) 3
4. b) Local minimum
5. a) 0
6. a) 2πr
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.
13. Refer to solution guide.
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
If the derivative changes from positive to negative, what does f have?
A local maximum, option (a) in this worksheet (Q1).
What is the slope of the tangent to y = x² at (1, 1)?
2, option (b) in this worksheet (Q2).
What does a sign change from negative to positive give?
A local minimum, from the answer key on this page.
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