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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
The volume of the spherical ball is increasing at the rate of 4π cc/sec. Find the rate at which the radius and the surface area are changing when the volume is 288 π cc.
Concept: Derivatives as a Rate Measure
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
Concept: Maxima and Minima
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
Concept: Maxima and Minima
The maximum value of the function f(x) = `logx/x` is ______.
Concept: Maxima and Minima
Find the equation of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`.
Concept: Applications of Derivatives in Geometry
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Concept: Increasing and Decreasing Functions
Find the approximate value of sin (30° 30′). Give that 1° = 0.0175c and cos 30° = 0.866
Concept: Approximations
Verify Lagrange’s mean value theorem for the function f(x) = `sqrt(x + 4)` on the interval [0, 5].
Concept: Lagrange's Mean Value Theorem (LMVT)
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
Concept: Maxima and Minima
Find the approximate value of tan−1 (1.002).
[Given: π = 3.1416]
Concept: Approximations
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Concept: Maxima and Minima
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Concept: Methods of Integration: Integration by Substitution
