In the following exercise, find the indefinite integral.

$$\int\left(e^t i+j+t \cos t k\right) d t$$

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = √t i + 5tj + 2t²k Time: t = 4

The figures show the paths of two particles.

Which vector, $\mathbf{s}$ or $\mathbf{t}$, represents the unit tangent vector? Explain.

In the following exercise, consider the vector-valued function $r(t)=3 t^2 i+(t-1) j+t k$. Write a vector-valued function u(t) that is the specified transformation of r.

A horizontal translation one unit in the direction of the positive x-axis

Find the indefinite integral. ∫ (sec² ti + 1/(1+t²) j) dt

In the following exercise, find the curvature and radius of curvature of the plane curve at the given value of x.

$$y=2 \sqrt{9-x^2}, \quad x=0$$

In the following exercise, use the model for projectile motion, assuming there is no air resistance and g = 9.8 meters per second per second.

A projectile is fired from ground level at an angle of $8^{\circ}$ with the horizontal. The projectile is to have a range of 50 meters. Find the minimum initial speed necessary.

The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector: r(t) = 4ti + t³j - tk Time: t = 1

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = 4ti + t³j - tk Time: t = 1

In the following exercise, consider the vector-valued function $r(t)=3 t^2 i+(t-1) j+t k$. Write a vector-valued function u(t) that is the specified transformation of r.

A vertical translation two units upward

An object moves along the path given by r(t) = 3ti + 4tj. Find v(t), a(t), T(t), and N(t) (if it exists). What is the form of the path? Is the speed of the object constant or changing?

In the following exercise, find the curvature and radius of curvature of the plane curve at the given value of x.

$$y=5 x^2+7, \quad x=-1$$

In the following exercise, find the indefinite integral.

$$\int\left(i+4 t^3 j+5^t k\right) d t$$

In the following exercise, use the model for projectile motion, assuming there is no air resistance and g = 9.8 meters per second per second.

Determine the maximum height and range of a projectile fired at a height of 1.5 meters above the ground with an initial speed of 100 meters per second and at angle of $30^{\circ}$ above the horizontal.