Online Test — Limits and Derivatives
10 Questions • 20 min • Chapter MCQ
20:00
Question 1 of 10
medium
If displacement s=t²+3t, then velocity at t=2 is:
5
7
9
11
Explanation: v=ds/dt=2t+3. At t=2, v=7.
Question 2 of 10
medium
If f(x)=2x³+x², then f'(x) is:
6x²+2x
6x+2
2x²+x
6x²+x
Explanation: Derivative=6x²+2x.
Question 3 of 10
easy
The slope of y=3x+2 is:
2
3
5
x
Explanation: The derivative of 3x+2 is 3.
Question 4 of 10
medium
If f(x)=2x²−3x+4, then f'(1) equals:
0
1
2
3
Explanation: f'(x)=4x−3. At x=1, value=1.
Question 5 of 10
medium
If f(x) = x³ − 3x, at which points does the tangent to the curve become horizontal?
x = 0 only
x = 1 and x = -1
x = 3 and x = -3
x = √3 and x = -√3
Explanation: Horizontal tangent means f'(x) = 0. f'(x) = 3x² − 3. Setting 3x² − 3 = 0 gives x² = 1, so x = 1 or -1.
Question 6 of 10
medium
The derivative of x⁷+2x⁴−1 is:
7x⁶+8x³
7x⁶+2x³
6x⁶+8x³
7x⁷+8x³
Explanation: Apply the power rule term-wise.
Question 7 of 10
medium
What is the derivative of f(x) = sec² x − tan² x?
2 sec x − 2 tan x
1
0
2 sec² x tan x
Explanation: By trigonometric identity, sec² x − tan² x = 1 for all valid x. The derivative of the constant 1 is 0.
Question 8 of 10
easy
lim(x→0) (sin7x)/(7x) equals:
0
1
7
49
Explanation: Standard limit directly gives 1.
Question 9 of 10
medium
Evaluate lim(x→0) (cosx − 1)/x².
−1
−1/2
0
1/2
Explanation: Since lim(x→0)(1−cosx)/x² = 1/2, this limit equals −1/2.
Question 10 of 10
medium
Evaluate lim(x→0) ((1 + x)² − 1)/x.
1
2
3
4
Explanation: Expand numerator: 1+2x+x²−1 = 2x+x². Limit = 2+x → 2.