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Federal (FBISE) · Chapter test · Free

2nd Year Mathematics Chapter 2 Test02: Limit, Continuity and Derivative

23 marks · 40 minutes · 14 questions. Covers: 02: Limit, Continuity and Derivative.

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2nd Year Mathematics Chapter 2 Test

2nd Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 23Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 3 × 1 = 3)

  1. If ff is differentiable for every x, then ff is
    1. Discontinuous
    2. Continuous
    3. inite
    4. Infinite
  2. If f(2)=2 f(2) = 2 , g(x)=x2 g(x) = x^2 , then ddx[3g(x)f(2)] \frac{d}{dx} \left[ \frac{3g(x)}{f(2)} \right]
    1. 2x
    2. 3x
    3. 32x\frac{3}{2}x
    4. 32x2\frac{3}{2}x^2
  3. If ff is differentiable for every value of x, then ff is continuous for:
    1. Some value of x
    2. [0 , ∞)
    3. Every value of x
    4. [−∞ , 0]

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Evaluate.lim⁡x→510x\lim_{x \to 5} 10x
  2. Use the concept of the differential to find the approximate value of the given expressions: (1.8)⁵
  3. Evaluate.lim⁡x→2(3x−2)6\lim_{x \to 2} \left(3x - 2 \right)^6
  4. Find dydx \frac{dy}{dx} if y=sin⁡x1+cos⁡x y = \frac{\sin x}{1 + \cos x}
  5. Use the second derivative to determine the intervals on which the function is concave upward and concave downward. f(x)=x(x−4)3f(x) = x(x - 4)^3
  6. Find the slope of the tangent line to the graph of the given function at the indicated point. f(x)=2x−1;(x,f(x))=(4,7)f(x) = 2x − 1; (x , f(x)) = (4 , 7)
  7. Find the slope of the tangent at x = 1, y = x⁴− x²
  8. Find the second derivative of the functions w.r.t. the variable involved: y = 2x⁶+ 5x³− 6x²

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. The side of a cube increases at a rate of 5 cm/hr. At what rate does the diagonal of the cube increase?
  2. Find dydxy=(x+1)2(x−1)2\frac{dy}{dx} y = \frac{(x+1)^2}{(x-1)^2}
  3. The height of a projectile shot from ground level is given by s(t) = −16t²+ 256t, where sss is measured in feet and t in seconds. (i) Determine the height of the projectile at t = 2, t = 6, t = 9, t = 10. (ii) What is the average velocity of the projectile between t = 2 and t = 5?