7. Antiderivatives & Indefinite Integrals

104–107. Functions from derivatives Find the function f with the following properties.

ƒ'(t) = sin t + 2t; ƒ(0) = 5

Solve the initial value problems in Exercises 55–58.

55. dy/dt = e^t sin(e^t − 2),y(ln 2) = 0

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

ds/dt = 1 + cos t, s(0) = 4

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

Solve the initial value problems in Exercises 87 and 88.

88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 3x⁻²ᐟ³, y(−1) = −5

Solve the initial value problems in Exercises 71–90.

y⁽⁴⁾ = −sin t + cos t;

y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0

Solve the following initial value problem:

$\(\frac{dy}{dx}\)=2x-5$; $y\(\left\)(0\(\right\))=4$

Solve the initial value problems in Exercises 53–56 for y as a function of x.

√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3

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) = 2 + 3 sin t; v(0) = 1, s(0) = 10

Find the function $f\(\left\)(x\(\right\))$ that satisfies the following differential equation.

$f^{\(\prime\]\prime\)}\(\left\)(x\(\right\))=3x^2$; $f^{\(\prime\)}\(\left\)(0\(\right\))=1$; $f\(\left\)(1\(\right\))=3$

Initial Value Problems

Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d²y/dx² = 2 − 6x; y′(0) = 4, y(0) = 1

In Exercises 5–8, show that each function is a solution of the given initial value problem.

7. Differential Equation: xy' + y = -sin(x), x>0

Initial condition: y(π/2) = 0

Solution candidate: y = cos(x)/x

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

a. Find the velocity of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.