Math Problem Solver

PART A – Continuity and Differentiability

Q1. Check the continuity of the following functions at the given point (any three):

(a) f(x) = (x2 - 9)/(x - 3), at x = 3

(b) f(x) = |x|, at x = 0

(c) f(x) = 1/x, at x = 0

(d) f(x) = { x2, if x <= 1 ; 2x + 1, if x > 1 }, at x = 1

Q2. Prove that the following functions are differentiable:

(a) f(x) = x3 + 2x2 + 5

(b) f(x) = sin(x) * cos(x)

Q3. Using the first principle of differentiation, find:

(a( d)/dx(x2)

(b) d/dx(√x)

Q4. If y = xx, find dy/dx.

PART B – Application of Derivatives

Q5. Find the equation of the tangent and normal to the curve y = x2 + 3x - 7 at x = 1.

Q6. A particle moves along a line with position s(t) = t3 - 6t2 + 9t + 5. Find:

(a) Velocity at time t

(b) Time when velocity is zero

Q7. Find the intervals where the function f(x) = 3x4 - 4x3 - 12x2 + 5 is increasing or decreasing.

Q8. A rectangular field is fenced on three sides with 120 m of fencing (one side along a wall). Find the dimensions that give the maximum area.

Additional Questions – Chapter 5

Q9. Find the value of k such that f(x) = {kx + 1, x <= 2 ; 3x - 1, x > 2} is continuous at x = 2.

Q10. If y = log(sin x), find dy/dx.

Q11. Differentiate y = tan-1(2x / (1 - x2)) with respect to x.

Q12. Show that f(x) = |x - 3| is not differentiable at x = 3.

Q13. If f(x) = ex cos(x), find f''(x)

Chapter 6

Q14. Find the maximum and minimum values of f(x) = x3 - 6x2 + 9x + 15.

Q15. A 10 m long ladder leans against a wall. The bottom slides away from the wall at 1 m/s. How fast is the top of the ladder sliding down when the bottom is 6 m from the wall?

Q16. Find the point on the curve y = x2 that is closest to the point (0, 3).

Q17. For y = (x2 + 1)/(x - 1), find intervals of monotonicity and identify local maxima and minima.

Q18. If C(x) = 3x3 - 10x2 + 100x + 500, find the marginal cost when x = 5.