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Find the approximate value of log10 (1016), given that log10e = 0⋅4343.
Concept: Approximations
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Concept: Maxima and Minima
Find the equation of the tangent to the curve at the point on it.
y = x2 + 2ex + 2 at (0, 4)
Concept: Applications of Derivatives in Geometry
A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?
Concept: Derivatives as a Rate Measure
Verify Lagrange’s mean value theorem for the following function:
f(x) = log x, on [1, e]
Concept: Lagrange's Mean Value Theorem (LMVT)
Divide the number 20 into two parts such that sum of their squares is minimum.
Concept: Maxima and Minima
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Concept: Maxima and Minima
Choose the correct option from the given alternatives:
Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.
Concept: Applications of Derivatives in Geometry
Choose the correct option from the given alternatives :
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.
Concept: Applications of Derivatives in Geometry
The approximate value of tan (44°30'), given that 1° = 0.0175c, is ______.
Concept: Approximations
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Concept: Maxima and Minima
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Concept: Maxima and Minima
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Concept: Maxima and Minima
The slope of the normal to the curve y = x2 + 2ex + 2 at (0, 4) is ______.
Concept: Applications of Derivatives in Geometry
If the tangent at (1, 1) on y2 = x(2 − x)2 meets the curve again at P, then P is
Concept: Applications of Derivatives in Geometry
The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. The time when the acceleration is 14 ft/sec2, is
Concept: Derivatives as a Rate Measure
A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______
Concept: Rolle's Theorem
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Concept: Increasing and Decreasing Functions
Water is being poured at the rate of 36 m3/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising
Concept: Derivatives as a Rate Measure
Find points on the curve given by y = x3 − 6x2 + x + 3, where the tangents are parallel to the line y = x + 5.
Concept: Applications of Derivatives in Geometry
