Limits, differentiation and integration

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 10 questions

Name: ____________________ Date: ____________

What is the limit of sin x / x as x approaches 0?
- A 0
- B It does not exist
- C 1
- D Infinity
What is d/dx of x³?
- A 3x³
- B 3x²
- C x⁴ / 4
- D x²
What is d/dx of e^(2x)?
- A 2e^(2x)
- B e^(2x)
- C e^(2x) / 2
- D 2x e^(2x)
What is the integral of x² dx?
- A x³ + C
- B x³ / 3 + C
- C 2x + C
- D x² / 2 + C
What is the value of the definite integral of 3x² from 0 to 2?
- A 12
- B 4
- C 8
- D 6
What is the minimum value of f(x) = x² - 4x + 5?
- A 1
- B -1
- C 2
- D 5
What is the limit of (x² - 4) / (x - 2) as x approaches 2?
- A 0
- B It does not exist
- C 4
- D 2
What is the slope of the tangent to y = x² at x = 3?
- A 9
- B 2
- C 6
- D 3
What is the integral of cos x dx?
- A sin x + C
- B -sin x + C
- C tan x + C
- D -cos x + C
What is d/dx of ln x?
- A e^x
- B x
- C 1/x
- D ln x

Answer key

1. C, 2. B, 3. A, 4. B, 5. C, 6. A, 7. C, 8. C, 9. A, 10. C

This is a standard limit and its value is 1 (with x in radians).
By the power rule, the derivative of x^n is n x^(n-1), so the answer is 3x².
By the chain rule, the derivative is e^(2x) multiplied by the derivative of 2x, which is 2.
The integral of x^n is x^(n+1) / (n+1) + C, so here it is x³ / 3 + C.
The antiderivative is x³. Evaluating gives 2³ - 0³ = 8.
Writing it as (x - 2)² + 1 shows the minimum is 1, at x = 2.
Factorise the numerator as (x - 2)(x + 2). Cancelling x - 2 leaves x + 2, which approaches 4.
dy/dx = 2x, which equals 6 at x = 3.
The derivative of sin x is cos x, so the integral of cos x is sin x + C.
The derivative of the natural logarithm is 1/x.