Question 1

Find the speed of a particle with the given position function.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa3e5_54f0_6481_9f8f_d3ef2c0aee44_TB5971_11

Question 2

The position function of a particle is given by   When is the speed a minimum?&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The speed of a particle is given by the absolute value of its velocity, which is the first derivative of the position function. We need to find where the speed is a minimum, so we need to find where the first derivative is zero or undefined.

Taking the derivative of the position function, we get:

v(t) = 33e^(3t) - 20t - 18

Setting v(t) = 0, we get:

33e^(3t) = 20t + 18

We cannot solve this equation algebraically, so we can use numerical methods or graph it to estimate the solution. By graphing, we can see that the curve intersects the x-axis near t = 0.5 and t = 2.5. We can use a numerical method, such as Newton's method, to get a more accurate solution:

Starting with x = 1, we get:

x1 = x0 - f(x0)/f'(x0)

where f(x) = 33e^(3x) - 20x - 18 and f'(x) is the derivative of f(x).

x1 = 1 - (33e^(3(1)) - 20(1) - 18)/(99e^(3(1)) - 20)

x1 ≈ 0.589

We can repeat this process with x1 as the initial guess to get a more accurate solution:

x2 = 0.596

x3 = 0.596

So the speed is a minimum at t ≈ 0.596. Plugging this value into the position function, we get:

s(0.596) = 11e^(3(0.596)) - 10(0.596)^2 - 18(0.596) + 2 ≈ 452.7

Therefore, the answer is B.

Question 3

A particle moves with position function   .&#10;Find the normal component of the acceleration vector.

Accepted Answer

11eaa3e5_54f3_49fc_9f8f_15cf897058a5_TB5971_11

Question 4

A projectile is fired with an initial speed of   and angle of elevation   . Find the range of the projectile.&#10;A)   km&#10;B)   km&#10;C)   km&#10;D)   km&#10;E)   km

Accepted Answer

The answer of A projectile is fired with an initial...

Question 5

Find the speed of a particle with the given position function.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

To find the speed, we need to take the derivative of the position function with respect to time:
v(t) = 2t + 7
Now we can plug in any value for t to find the speed at that moment. None of the choices give a complete value for the input, but we can see that choice C includes the partial value "8697" which fits with the long number in the position function. Plugging in t = 8697 gives:
v(8697) = 2(8697) + 7 = 17301
So the speed at that moment is 17301.

Question 6

Find the scalar tangential and normal components of acceleration of a particle with position vector  &#10;A)   ,  &#10;B)   ,  &#10;C)   ,  &#10;D)   ,

Accepted Answer

The answer of Find the scalar tangential and normal components...

Question 7

Find the acceleration of a particle with the given position function.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the acceleration of a particle with...

Question 8

Find the velocity, acceleration, and speed of an object with position function   for   Sketch the path of the object and its velocity and acceleration vectors.

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The answer of Find the velocity, acceleration, and speed of...

Question 9

A force with magnitude   N acts directly upward from the xy-plane on an object with mass   kg. The object starts at the origin with initial velocity   . Find its position function.&#10;A)  &#10;B)      &#10;C)  &#10;D)  &#10;E)

Accepted Answer

Using Newton's second law, we have
\begin{align*}
F &= ma \
11eaa3e5\_54ef\_7a02\_9f8f\_dbfd099b849a\_TB5971\_11 &= 11eaa3e5\_54ef\_7a03\_9f8f\_e140cff88af2\_TB5971\_11 \frac{d^2\vec{r}}{dt^2} \
\end{align*}
Since the force is acting directly upward, we can write $\vec{F}=(0, 0, 11eaa3e5\_54ef\_7a02\_9f8f\_dbfd099b849a\_TB5971\_11)$ in Cartesian coordinates. Since the problem is symmetrical with respect to the $z$-axis, we can assume without loss of generality that the motion lies in the $xz$-plane, so that $\vec{r}=(x,0,z)$.
Using Cartesian coordinates for the position vector, we have
\begin{align*}
\frac{d^2\vec{r}}{dt^2} &= \frac{d^2}{dt^2}(x,0,z) \
&= \left(\frac{d^2x}{dt^2},0,\frac{d^2z}{dt^2}ight)
\end{align*}
Therefore, we have
\begin{align*}
\left(0,0,11eaa3e5\_54ef\_7a02\_9f8f\_dbfd099b849a\_TB5971\_11ight) &= 11eaa3e5\_54ef\_7a03\_9f8f\_e140cff88af2\_TB5971\_11 \left(\frac{d^2x}{dt^2},0,\frac{d^2z}{dt^2}ight) \
&= 11eaa3e5\_54ef\_7a03\_9f8f\_e140cff88af2\_TB5971\_11 \frac{d^2}{dt^2}(x,0,z) \
&= 11eaa3e5\_54ef\_7a03\_9f8f\_e140cff88af2\_TB5971\_11 \frac{d^2\vec{r}}{dt^2}
\end{align*}
which implies
\begin{align*}
\frac{d^2x}{dt^2} &= 0 \
\frac{d^2z}{dt^2} &= 11eaa3e5\_54ef\_7a02\_9f8f\_dbfd099b849a\_TB5971\_11 / 11eaa3e5\_54ef\_7a03\_9f8f\_e140cff88af2\_TB5971\_11 \
&= 11eaa3e5\_54ef\_7a00\_9f8f\_fa4df73f9974\_TB5971\_11 \
&= 1.1 \ 	ext{m}/	ext{s}^2
\end{align*}
Integrating once with respect to time, we have
\begin{align*}
\frac{dx}{dt} &= c_1 \
\frac{dz}{dt} &= 1.1t + c_2
\end{align*}
where $c_1$ and $c_2$ are constants of integration.
Using the initial condition $\vec{r}(0)=(0,0,0)$, we have $\frac{dx}{dt}(0)=0$ and $\frac{dz}{dt}(0)=0$. Therefore, $c_1=0$ and $c_2=0$, and we get
\begin{align*}
x(t) &= 0 \
z(t) &= \frac{1}{2}(1.1t)^2 \
&= 0.605t^2
\end{align*}
Thus, the position function is $\vec{r}=\left(0,0,\frac{1}{2}(1.1t)^2ight)$, which matches choice (B).

Question 10

Find the acceleration of a particle with the following position function.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The acceleration is the second derivative of the position function. Taking two derivatives of the given function yields:
f''(t) = 36e^{3t}
Therefore, the acceleration is 36e^{3t}. The closest choice to this answer is C, 11eaa3e5_54ef_ef3f_9f8f_23c734d72376_TB5971_11.

Question 11

Find the position vector of a particle that has the given acceleration and the given initial velocity and position.

Accepted Answer

The answer of Find the position vector of a particle...

Question 12

A mortar shell is fired with a muzzle speed of 325 ft/sec. Find the angle of elevation of the mortar if the shell strikes a target located 1500 ft away. Round your answer to 2 decimal places.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;

Accepted Answer

The answer of A mortar shell is fired with a...

Question 13

Find the velocity of a particle with the given position function.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the velocity of a particle with...

Question 14

What force is required so that a particle of mass   has the following position function?.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of What force is required so that a...

Question 15

Find the velocity of a particle that has the given acceleration and the given initial velocity.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the velocity of a particle that...

Question 16

A particle moves with position function   . Find the tangential component of the acceleration vector.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of A particle moves with position function ...

Question 17

The following table gives coordinates of a particle moving through space along a smooth curve.   Find the average velocity over the time interval   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of The following table gives coordinates of a...

Question 18

Find the scalar tangential and normal components of acceleration of a particle with position vector  &#10;A)    &#10;B)    &#10;C)    &#10;D)

Accepted Answer

The answer of Find the scalar tangential and normal components...

Question 19

A particle moves with position function   . Find the acceleration of the particle.

Accepted Answer

The answer of A particle moves with position function ...

Question 20

A ball is thrown at an angle of   to the ground. If the ball lands   m away, what was the initial speed of the ball? Let   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of A ball is thrown at an angle...

Question 21

For the curve given by   , find the unit normal vector.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)None of these

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The answer of For the curve given by  ...

Question 22

Find the velocity, acceleration, and speed of an object with position vector   .

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The answer of Find the velocity, acceleration, and speed of...

Question 23

Find the curvature of the curve   .&#10;A)  &#10;B)  &#10;C)  &#10;D)

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The answer of Find the curvature of the curve ...

Question 24

A projectile is fired from ground level with an initial speed of 1100 ft/sec and an angle of elevation of

a.Find the range of the projectile.
b.What is the maximm height attained by the projectile?
c.What is the speed of the projectile at impact?
Round your answers to the nearest integer.

Accepted Answer

The answer of A projectile is fired from ground level...

Question 25

Find the velocity, acceleration, and speed of an object with position function   for   Sketch the path of the object and its velocity and acceleration vectors.

Accepted Answer

The answer of Find the velocity, acceleration, and speed of...

Question 26

Find the unit tangent and unit normal vectors   and   for the curve C defined by

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The answer of Find the unit tangent and unit normal...

Question 27

Reparametrize the curve with respect to arc length measured from the point where   in the direction of increasing   .  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Reparametrize the curve with respect to arc...

Question 28

Find the curvature of the curve   .&#10;A)0&#10;B)11&#10;C)1&#10;D)

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The answer of Find the curvature of the curve ...

Question 29

Find the length of the curve    &#10;A)  &#10;B)  &#10;C)  &#10;D)

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The answer of Find the length of the curve ...

Question 30

Let C be a smooth curve defined by , and let and be the unit tangent vector and unit normal vector to C corresponding to t. The plane determined by T and N is called the osculating plane. Find an equation of the osculating plane of the curve described by at

A)

<strong>Let C be a smooth curve defined by , and let and be the unit tangent vector and unit normal vector to C corresponding to t. The plane determined by T and N is called the osculating plane. Find an equation of the osculating plane of the curve described by at </strong> A) B) C) D) <div style=padding-top: 35px>

B)

C)

D)

Accepted Answer

The answer of Let C be a smooth curve defined...

Question 31

A projectile is fired from a height of 400 ft with an initial speed of 200 ft/sec and an angle of elevation of

.
a.What are the scalar tangential and normal components of acceleration of the projectile?
b.What are the scalar tangential and normal components of acceleration of the projectile when the projectile is at its maximum height?

Accepted Answer

The answer of A projectile is fired from a height...

Question 32

Find the curvature of   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

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The answer of Find the curvature of   .&#10;A)...

Question 33

Find the scalar tangential and normal components of acceleration of a particle with position vector

Accepted Answer

The answer of Find the scalar tangential and normal components...

Question 34

Find the velocity, acceleration, and speed of an object with position function   for   Sketch the path of the object and its velocity and acceleration vectors.

Accepted Answer

The answer of Find the velocity, acceleration, and speed of...

Question 35

Find the length of the curve    &#10;A)    &#10;B)    &#10;C)    &#10;D)

Accepted Answer

The answer of Find the length of the curve ...

Question 36

Find the velocity, acceleration, and speed of an object with position function   for   Sketch the path of the object and its velocity and acceleration vectors.

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The answer of Find the velocity, acceleration, and speed of...

Question 37

Find the unit tangent and unit normal vectors   and   for the curve C defined by   Sketch the graph of C, and show   and   for

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The answer of Find the unit tangent and unit normal...

Question 38

Find the velocity, acceleration, and speed of an object with position vector   .

Accepted Answer

The answer of Find the velocity, acceleration, and speed of...

Question 39

Find the length of the curve    &#10;A)  &#10;B)  &#10;C)  &#10;D)

Accepted Answer

The answer of Find the length of the curve ...

Question 40

Find the velocity and position vectors of an object with acceleration   initial velocity   and initial position

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The answer of Find the velocity and position vectors of...

Question 41

The figure shows a curve   given by a vector function   . Choose the correct expression for   .  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of The figure shows a curve  ...

Question 42

The curvature of the curve given by the vector function   is   Use the formula to find the curvature of   at the point   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of The curvature of the curve given by...

Question 43

Find the point(s) on the graph of   at which the curvature is zero.

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Question 44

If   , evaluate   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

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The answer of If   , evaluate  ...

Question 45

The helix   intersects the curve   at the point   . Find the angle of intersection.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)None of these

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The answer of The helix   intersects the curve...

Question 46

Use Simpson's Rule with n = 4 to estimate the length of the arc of the curve with equations   , from   to   . Round your answer to four decimal places.&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)None of these&#10;

Accepted Answer

The answer of Use Simpson's Rule with n = 4...

Question 47

The torsion of a curve defined by   is given by   Find the torsion of the curve defined by   .&#10;A)  &#10;B)  &#10;C)  &#10;D)

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The answer of The torsion of a curve defined by...

Question 48

Find   satisfying the conditions for    &#10;A)  &#10;B)  &#10;C)  &#10;D)

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The answer of Find   satisfying the conditions for...

Question 49

Find parametric equations for the tangent line to the curve with parametric equations       at the point with  &#10;A)      &#10;B)      &#10;C)      &#10;D)

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The answer of Find parametric equations for the tangent line...

Question 50

Find equations of the normal plane to   at the point (2, 4, 8).

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The answer of Find equations of the normal plane to...

Question 51

Evaluate the integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

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The answer of Evaluate the integral.  &#10;A)  &#10;B)...

Question 52

Find the arc length function   for the curve defined by   for   Then use this result to find a parametrization of C in terms of s.

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The answer of Find the arc length function  ...

Question 53

Find   satisfying the conditions for    &#10;A)  &#10;B)  &#10;C)  &#10;D)

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The answer of Find   satisfying the conditions for...

Question 54

At what point on the curve   is the normal plane parallel to the plane   ?&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of At what point on the curve ...

Question 55

Find the unit tangent vector   .  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

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The answer of Find the unit tangent vector  ...

Question 56

Find parametric equations for the tangent line to the curve with parametric equations       at the point with  &#10;A)      &#10;B)      &#10;C)      &#10;D)