Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = -2x⁴ + x² + 10

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.

v(t) = 2√t; s(0) = 1

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = sin x + x - 1; x₀ = 0.5

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]

Rectangles beneath a semicircle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

Solving initial value problems Find the solution of the following initial value problems.

y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ); y (π/4) = 3, -π/2 < Θ < π/2