You CAN Ace Calculus

other exams from this semester

exam A1 - exam A2 - exam A3 - exam A4

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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This is the first exam for third semester, multi-variable calculus. The exam covers the basics of vectors and vector functions and applications including curvature, arc length and tangent and normal vectors.

### Practice Exam Tips

Each exam page contains a full exam with detailed solutions. Most of these are actual exams from previous semesters used in college courses. You may use these as practice problems or as practice exams. Here are some suggestions on how to use these to help you prepare for your exams.

- Set aside a chunk of full, uninterrupted time, usually an hour to two, to work each exam.
- Go to a quiet place where you will not be interrupted that duplicates your exam situation as closely as possible.
- Use the same materials that you are allowed in your exam (unless the instructions with these exams are more strict).
- Use your calculator as little as possible except for graphing and checking your calculations.
- Work the entire exam before checking any solutions.
- After checking your work, rework any problems you missed and go to the 17calculus page discussing the material to perfect your skills.
- Work as many practice exams as you have time for. This will give you practice in important techniques, experience in different types of exam problems that you may see on your own exam and help you understand the material better by showing you what you need to study.

IMPORTANT -
Exams can cover only so much material. Instructors will sometimes change exams from one semester to the next to adapt an exam to each class depending on how the class performs during the semester while they are learning the material. So just because you do well (or not) on these practice exams, does not necessarily mean you will do the same on your exam. Your instructor may ask completely different questions from these. That is why working lots of practice problems will prepare you better than working just one or two practice exams.
Calculus is not something that can be learned by reading. You have to work problems on your own and struggle through the material to really know calculus, do well on your exam and be able to use it in the future.

Exam Details

Time

2 hours

Questions

5

Total Points

75

Tools

Calculator

no

Formula Sheet(s)

1 page, 8.5x11 or A4

Other Tools

none

Instructions:
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

(20 points) For $$\vec{r}(t) = 2\cos(t^2) \hat{i} + 2\sin(t^2)\hat{j}$$, $$t \ge 0$$, compute
a. $$\vec{T}(t)$$; b. $$\vec{N}(t)$$; c. curvature; d. $$a_T$$ and $$a_N$$; e. $$\vec{r}(s)$$

Problem Statement

(20 points) For $$\vec{r}(t) = 2\cos(t^2) \hat{i} + 2\sin(t^2)\hat{j}$$, $$t \ge 0$$, compute
a. $$\vec{T}(t)$$; b. $$\vec{N}(t)$$; c. curvature; d. $$a_T$$ and $$a_N$$; e. $$\vec{r}(s)$$

Solution

a. $$\displaystyle{\vec{T} = \frac{\vec{v}(t)}{\|\vec{v}(t)\|}}$$
$$\vec{r}'(t)=\vec{v}(t)=-2\sin(t^2)(2t)\hat{i}+2\cos(t^2)(2t)\hat{j}$$
$$\|\vec{v}(t)\| = \sqrt{ [4t\sin(t^2)]^2+[4t\cos(t^2)]^2} =$$ $$\sqrt{ 16t^2\sin^2(t^2) + 16t^2\cos^2(t^2) } =$$ $$\sqrt{ 16t^2[\sin^2(t^2) + \cos^2(t^2)]} = \sqrt{16t^2} = 4t$$
$$\displaystyle{ \vec{T} = \frac{\vec{v}(t)}{\|\vec{v}(t)\|} = \frac{-4t\sin(t^2)\hat{i}+4t\cos(t^2)\hat{j}}{4t}}$$
$$\boxed{ \vec{T} = -\sin(t^2)\hat{i}+\cos(t^2)\hat{j}}$$

b. $$\displaystyle{\vec{N}(t) = \frac{ d\vec{T}/dt }{ \|d\vec{T}/dt\|}}$$
$$\displaystyle{ \frac{d\vec{T}}{dt} = -\cos(t^2)(2t)\hat{i} + (-\sin(t^2))(2t)\hat{j}}$$
$$\| d\vec{T}/dt \| = \sqrt{ 4t^2\cos^2(t^2) + 4t^2\sin^2(t^2)} =$$ $$\sqrt{4t^2} = 2t$$
$$\displaystyle{\vec{N}(t) = \frac{-2t\cos(t^2)\hat{i}-2t\sin(t^2)\hat{j}}{2t}}$$
$$\boxed{ \vec{N}(t) = -\cos(t^2)\hat{i} -\sin(t^2)\hat{j}}$$

c. $$\displaystyle{ K=\frac{1}{\|\vec{v}\|} \left\|\frac{d\vec{T}}{dt}\right\| = }$$ $$\displaystyle{\frac{1}{4t}(2t) = \frac{1}{2}}$$     $$\boxed{K=1/2}$$

d. $$a_N = k \|\vec{v}\|^2 = (1/2)(4t)^2 = 8t^2$$     $$\boxed{a_N=8t^2}$$
You may be tempted to use the equation $$\displaystyle{a_T = \frac{\vec{v} \cdot \vec{a}}{\|\vec{v}\|}}$$, which will work but it is the hard way. However, if you look ahead at the next part of this problem, you are asked to find $$\vec{r}(s)$$, the arc length parameterization of $$\vec{r}(t)$$. During the course of solving that, you will find that $$s=t^2$$. So, to find $$a_T$$, we can use $$\displaystyle{a_T=\frac{d^2s}{dt^2}}$$.
$$\displaystyle{a_T=\frac{d^2s}{dt^2} = \frac{d}{dt}[4t] = 4}$$    $$\boxed{a_T=4}$$

e. $$\displaystyle{s(t) = \int_{a}^{t}{\|\vec{v}(u)\|~du}}$$
Since $$\|\vec{v}(u)\| = 4u$$, we have $$\displaystyle{ s(t)=\int_{0}^{t}{4u~du} = \left. 2u^2\right|_0^t = 2t^2}$$
So $$s=2t^2$$. Solving for t, we have $$s=2t^2 \to s/2=t^2 \to t=\pm\sqrt{s/2}$$. Since $$t\ge0$$, we choose the positive square root, giving us $$t=\sqrt{s/2}$$. Substituting for t in the original position function, we get $$\vec{r}$$ in terms of s.
$$\boxed{\vec{r}(s) = 2\cos(s/2)\hat{i} + 2\sin(s/2)\hat{j} ~~~ s \ge 0}$$

(5 points) Compute the indefinite integral of $$\vec{r}(t)= (5t^{-4}-t^2)\hat{i} + (t^6-4t^3)\hat{j} + (2/t)\hat{k}$$

Problem Statement

(5 points) Compute the indefinite integral of $$\vec{r}(t)= (5t^{-4}-t^2)\hat{i} + (t^6-4t^3)\hat{j} + (2/t)\hat{k}$$

$$\displaystyle{ \int{ \vec{r}(t)~dt } = \left( \frac{-5}{3}t^{-3}-\frac{t^3}{3} \right)\hat{i} + \left( \frac{t^7}{7}-t^4 \right) \hat{j} + \left( 2\ln \abs{t} \right)\hat{k} + \vec{C} }$$

Problem Statement

(5 points) Compute the indefinite integral of $$\vec{r}(t)= (5t^{-4}-t^2)\hat{i} + (t^6-4t^3)\hat{j} + (2/t)\hat{k}$$

Solution

Evaluating each integral separately, gives us
x-component: $$\displaystyle{\int{ 5t^{-4}-t^2~dt } = \frac{5t^{-3}}{-3} - \frac{t^3}{3} + c_1}$$
y-component: $$\displaystyle{\int{ t^6-4t^3 ~dt} = \frac{t^7}{7}-\frac{4t^4}{4} + c_2}$$
z-component: $$\displaystyle{\int{ \frac{2}{t}~dt} = 2\ln \abs{t} + c_3}$$
Let $$\vec{C}= c_1\hat{i} + c_2 \hat{j} + c_3 \hat{k}$$.

$$\displaystyle{ \int{ \vec{r}(t)~dt } = \left( \frac{-5}{3}t^{-3}-\frac{t^3}{3} \right)\hat{i} + \left( \frac{t^7}{7}-t^4 \right) \hat{j} + \left( 2\ln \abs{t} \right)\hat{k} + \vec{C} }$$

(10 points) For the trajectory $$\vec{r}(t) = \langle e^t\sin(t), e^t\cos(t), e^t\rangle$$, find
a. the speed associated with the trajectory;
b. the length of the trajectory on the interval $$0\leq t \leq \ln(2)$$

Problem Statement

(10 points) For the trajectory $$\vec{r}(t) = \langle e^t\sin(t), e^t\cos(t), e^t\rangle$$, find
a. the speed associated with the trajectory;
b. the length of the trajectory on the interval $$0\leq t \leq \ln(2)$$

Solution

a. speed $$\|\vec{v}(t)\|$$ $$\vec{v}(t) = \vec{r}'(t) = \langle e^t\cos(t)+e^t\sin(t), -e^t\sin(t)+e^t\cos(t), e^t \rangle$$
$$\|\vec{v}(t)\| = \sqrt{ (e^t\cos t + e^t \sin t)^2 + (-e^t\sin t+e^t\cos t)^2 + e^{2t} } =$$
$$e^t\sqrt{ \cos^2 t + 2\sin t \cos t + \sin^2 t + \sin^2 t - 2\sin t \cos t + \cos^2 t + 1 } =$$
$$e^t \sqrt{1+1+1}$$
$$\boxed{ \|\vec{v}(t) \| = e^t\sqrt{3}}$$

b. $$\displaystyle{L = \int_{a}^{b}{ \|\vec{v}(t)\| } = \int_{0}^{\ln 2}{ e^t\sqrt{3}~dt} = }$$
$$\displaystyle{\left. e^t\sqrt{3}\right|_{0}^{\ln 2} = }$$
$$\sqrt{3}e^{\ln 2} - \sqrt{3}e^0 = 2\sqrt{3}-\sqrt{3} = \sqrt{3}$$
$$\boxed{L=\sqrt{3}}$$

(15 points) For the vectors $$\vec{u}=\langle 1,3,-2 \rangle$$ and $$\vec{v}=\langle 4,2,1 \rangle$$, find
a. the cosine of the angle $$\theta$$ between the vectors
b. the projection of $$\vec{v}$$ onto $$\vec{u}$$; c. $$\vec{u} \times \vec{v}$$

Problem Statement

(15 points) For the vectors $$\vec{u}=\langle 1,3,-2 \rangle$$ and $$\vec{v}=\langle 4,2,1 \rangle$$, find
a. the cosine of the angle $$\theta$$ between the vectors
b. the projection of $$\vec{v}$$ onto $$\vec{u}$$; c. $$\vec{u} \times \vec{v}$$

Solution

a. To find the cosine of the angle, use $$\vec{u}\cdot\vec{v}= \|\vec{u}\|~\|\vec{v}\| \cos \theta$$
$$\vec{u}\cdot\vec{v}=1(4)+3(2)+(-2)(1)=4+6-2 = 8$$
$$\|\vec{u}\| = \sqrt{1^2+3^2+(-2)^2} = \sqrt{14}$$
$$\|\vec{v}\| = \sqrt{4^2+2^2+1^2} = \sqrt{21}$$
$$\displaystyle{ \cos \theta = \frac{8}{\sqrt{14}\sqrt{21}}}$$     $$\boxed{ \displaystyle{ \cos \theta = \frac{8}{7\sqrt{6}}}}$$

b. $$\displaystyle{proj_{\vec{u}}\vec{v} = \|\vec{v}\| \cos\theta\left( \frac{\vec{u}}{\|\vec{u}\|} \right) = }$$ $$\displaystyle{\sqrt{21}\frac{8}{7\sqrt{6}} \frac{\langle1,3,-2\rangle}{\sqrt{14}} = }$$ $$\displaystyle{\frac{4}{7}\langle 1,3,-2 \rangle}$$

c. $$\vec{u}\times\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & -2 \\ 4 & 2 & 1 \end{vmatrix} =$$ $$\displaystyle{\hat{i}(3+4) - \hat{j}(1+8) + \hat{k}(2-12)}$$
$$\boxed{\vec{u}\times\vec{v}=\langle 7,-9,-10\rangle}$$

(25 points) A baseball is hit 3 ft above home plate with an initial velocity of $$\langle 65,85,85\rangle$$ ft/sec. The spin on the baseball produces a horizontal acceleration of the ball of 12 ft/sec2 in the eastward direction. Assume the x-axis points east, the y-axis points north and the z-axis is vertical with g=32 ft/sec2.
a. Find the velocity vector for $$t\ge0$$.     b. Find the position vector for $$t\ge0$$.
c. Clearly explain how you would determine the time of flight and range of the ball (do not calculate).
d. Clearly explain how you would determine the maximum height of the ball (do not calculate).

Problem Statement

(25 points) A baseball is hit 3 ft above home plate with an initial velocity of $$\langle 65,85,85\rangle$$ ft/sec. The spin on the baseball produces a horizontal acceleration of the ball of 12 ft/sec2 in the eastward direction. Assume the x-axis points east, the y-axis points north and the z-axis is vertical with g=32 ft/sec2.
a. Find the velocity vector for $$t\ge0$$.     b. Find the position vector for $$t\ge0$$.
c. Clearly explain how you would determine the time of flight and range of the ball (do not calculate).
d. Clearly explain how you would determine the maximum height of the ball (do not calculate).

Solution

a. $$\vec{a}=\langle 12,0,-32\rangle$$ ft/sec2
$$\vec{v}=\int{\vec{a}~dt}=\langle 12t+c_1, c_2, -32t+c_3 \rangle$$
$$\vec{v}(0)=\langle c_1,c_2,c_3 \rangle = \langle 65,85,85 \rangle$$
$$\boxed{ \vec{v}(t)=\langle 12t+65, 85, -32t+85 \rangle ~ ft/sec }$$

b. $$\vec{r}(t) = \int{\vec{v}~dt} =$$ $$\langle 6t^2+65t+k_1, 85t+k_2, -16t^2+85t+k_3 \rangle$$
$$\vec{r}(0) = \langle k_1, k_2, k_3 \rangle = \langle 0,0,3 \rangle$$
$$\boxed{\vec{r}(t)=\langle 6t^2+65t, 85t, -16t^2+85t+3 \rangle ~ ft }$$

c. To calculate the time of flight, set the z-component of the position vector to zero an solve for t. You will get two values since the equation is a quadratic. The small value is negative, so it has no significance. The large value will be the time when the ball hits the ground, which is the time of flight.
To find the range, plug the value of t found for the time of flight into the position and find the magnitude of that vector to get the range.

d. To calculate the maximum height, we take the z-component of the velocity vector and set it equal to zero. This is equivalent to taking the derivative of the position of vector and set it equal to zero that you are familiar with from calculus 1 to find the critical value. Solving for t gives us the time when the ball is at it's maximum value. We take that value of t and plug it into the z-component of the position vector to get the maximum height.