7. Antiderivatives & Indefinite Integrals

Using the acceleration function below, find the velocity function, if the velocity is v = 5 at time t = 2.

$a\(\left\)(t\(\right\))=-20$

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$

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.

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 .

d. Find the time when the object strikes the ground.

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.

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 .

b. Find the position 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.

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 .

c. Find the time when the object reaches its highest point. What is the height? 

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.

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) = 2t + 4; s(0) = 0

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

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

g'(x) = 7x(x⁶ - 1/7); g(1) = 2

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

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) = 6t² + 4t - 10; s(0) = 0

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

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

A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?

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

ƒ'(t) = sin t + 2t; ƒ(0) = 5