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05:44{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
f'(x) = 3x + sinx; f(0) = 3
{Use of Tech} Estimating roots The values of various roots can be approximated using Newton’s method. For example, to approximate the value of ³√10, we let x = ³√10 and cube both sides of the equation to obtain x³ = 10, or x³ - 10 = 0. Therefore, ³√10 is a root of p(x) = x³ - 10, which we can approximate by applying Newton’s method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of x₀ and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding.
r = 7¹/⁴
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = eˣ/(e²ᵉ + 1)
Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = t/ t² + 1
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = -32; v(0) = 20, s(0) = 0
