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#### Application of Differential Calculus - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:30:00 Hrs
Total Marks : 30
15 x 2 = 30
1. A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) =100×(1− 0.1t)2, 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

2. A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres
Find the instantaneous velocities at t = 3 and t = 6 seconds.

3. A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t2 in t seconds
What is the instantaneous velocity of the camera when it hits the ground?

4. Compute the value of 'c' satisfied by the Rolle’s theorem for the function f (x) = x2 (1 - x)2, x ∈ [0.1]

5. Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
$f(x)=|\frac{1}{x}|, x\in [-1,1]$

6. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
$f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]$

7. Find two positive numbers whose sum is 12 and their product is maximum.

8. The volume of a cylinder is given by the formula V = πr2 h. Find the greatest and least values of V if r + h = 6.

9. Evaluate the following limit, if necessary use l’Hôpital Rule
$\underset { x\rightarrow \infty }{ lim } \frac { { 2x }^{ 2 }-3 }{ { x }^{ 2 }-5x+3 }$

10. Prove that the function f (x) = x2 − 2x − 3 is strictly increasing in $(2, \infty)$

11. Find the slant (oblique) asymptote for the function $f(x)=\frac { { x }^{ 2 }-6x+7 }{ x+5 }$

12. Find the point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa.

13. Find the point at which the curve y-exy+x=0 has a vertical tangent.

14. Verify Rolle ’s Theorem for $f(x)=\left| x-1 \right| ,O\le x\le 2$

15. Expand the polynomial f(x)=x2-3x+2 in power of (x-2)