7. Antiderivatives & Indefinite Integrals

Solve the initial value problem: $x^{\prime }=2x$, $x\left(0\right)=4$. What is $x\left(t\right)$?

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

Consider the initial value problem: $y^{\mathrm{\text{'}\text{'}}}+16y=\cos \left(4t\right)$, with $y\left(0\right)=0$ and $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is the particular solution $y\left(t\right)$?

Solve the following initial value problem:

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

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