You CAN Ace Calculus

### Topics You Need To Understand For This Page

 vectors vector functions

### Related Topics and Links

unit tangent vector youtube playlist

### Calculus Topics 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

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

In order to discuss curvature and a few other topics, we need to define a special vector called the unit tangent vector. As the name indicates, the unit tangent vector is a vector that is tangent to the curve and it's length is one.

What may not be obvious is that there is only one unit tangent vector and it points in the direction of motion. Given the vector function $$\vec{r}(t)$$, the most basic equation we use to find the unit tangent vector is

$$\displaystyle{ \vec{T}(t) = \frac{\vec{r}'(t)}{ \| \vec{r}'(t) \| } }$$

The vector function $$\vec{r}(t)$$ is often a position vector. As you know from basic calculus, the derivative of the position is velocity. So you will often see $$\vec{v}(t)=\vec{r}'(t)$$ where $$\vec{v}(t)$$ is referred to as the velocity vector. This allows us to write the unit tangent vector as $$\displaystyle{ \vec{T}(t) = \frac{\vec{v}(t)}{ \| \vec{v}(t) \| } }$$.

This unit tangent vector is used a lot when calculating the principal unit normal vector, acceleration vector components and curvature. So take a few minutes to work some practice problems before going on to the next topic.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, calculate the unit tangent vector for the given vector function at the given point. If no point is given, find the general unit tangent vector $$\vec{T}(t)$$.

$$\vec{r}(t) = t\vhat{i} + (1/t)\vhat{j}$$, $$t=1$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = t\vhat{i} + (1/t)\vhat{j}$$ at the point $$t=1$$.

$$\displaystyle{ \vec{T}(1) = \frac{\vhat{i} - \vhat{j}}{\sqrt{2}} }$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = t\vhat{i} + (1/t)\vhat{j}$$ at the point $$t=1$$.

Solution

$$\vec{r}'(t) = \vhat{i} - (1/t^2)\vhat{j}$$
$$\vec{r}'(1) = \vhat{i} - \vhat{j}$$
$$\displaystyle{ \vec{T}(1) = \frac{\vec{r}'(1)}{\| \vec{r}'(1) \|} = \frac{\vhat{i}-\vhat{j}}{\sqrt{2}} }$$
Here is a plot of the solution. The black line is the curve $$\vec{r}(t)$$ with the black arrow indicating that it is being traced out left to right (or down the curve). (We have plotted only a section of the curve from x=1/4 to x=4.) The red vector is the unit tangent vector. Notice that it is pointing in the direction that the curve is being traced.

$$\displaystyle{ \vec{T}(1) = \frac{\vhat{i} - \vhat{j}}{\sqrt{2}} }$$

$$\vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} +$$ $$2\sin 2t \vhat{k}$$, $$t=0$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} +$$ $$2\sin 2t \vhat{k}$$ at the point $$t=0$$.

$$\vec{T}(0) = (3/5)\vhat{j} + (4/5)\vhat{k}$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \cos t \vhat{i} + 3t\vhat{j} +$$ $$2\sin 2t \vhat{k}$$ at the point $$t=0$$.

Solution

### 2048 video

video by Krista King Math

$$\vec{T}(0) = (3/5)\vhat{j} + (4/5)\vhat{k}$$

$$\vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle, t=\pi/6$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle$$ at the point $$t=\pi/6$$.

$$\displaystyle{ \frac{1}{\sqrt{19}} \langle \sqrt{3}, -2, 2\sqrt{3} \rangle }$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle 2\sin(t), 4\cos(t), 4\sin^2(t) \rangle$$ at the point $$t=\pi/6$$.

Solution

### 2050 video

video by MIP4U

$$\displaystyle{ \frac{1}{\sqrt{19}} \langle \sqrt{3}, -2, 2\sqrt{3} \rangle }$$

$$\vec{r}(t) = \langle t^3,2t^2 \rangle, t=1$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle t^3,2t^2 \rangle$$ at the point $$t=1$$.

$$\langle 3/5,4/5 \rangle$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle t^3,2t^2 \rangle$$ at the point $$t=1$$.

Solution

### 2049 video

video by MIP4U

$$\langle 3/5,4/5 \rangle$$

Find $$\vec{T}(t)$$ and $$\vec{T}(0)$$ for $$\vec{r}(t)=\langle 5t^2+1, -e^{-3t}, 2\sin(-3t) \rangle$$.

Problem Statement

Find $$\vec{T}(t)$$ and $$\vec{T}(0)$$ for $$\vec{r}(t)=\langle 5t^2+1, -e^{-3t}, 2\sin(-3t) \rangle$$.

$$\displaystyle{ \vec{T}(t) = \frac{\langle 10t, 3e^{-3t}, -6\cos(-3t) \rangle}{\sqrt{100t^2+9e^{-6t}+36\cos^2(-3t)}} }$$
$$\vec{T}(0) = \langle 0,1/\sqrt{5},-2/\sqrt{5} \rangle$$

Problem Statement

Find $$\vec{T}(t)$$ and $$\vec{T}(0)$$ for $$\vec{r}(t)=\langle 5t^2+1, -e^{-3t}, 2\sin(-3t) \rangle$$.

Solution

### 2051 video

video by MIP4U

$$\displaystyle{ \vec{T}(t) = \frac{\langle 10t, 3e^{-3t}, -6\cos(-3t) \rangle}{\sqrt{100t^2+9e^{-6t}+36\cos^2(-3t)}} }$$
$$\vec{T}(0) = \langle 0,1/\sqrt{5},-2/\sqrt{5} \rangle$$

$$\vec{r}(t) = (-t^3+t)\vhat{i} +$$ $$(\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k}$$, $$t=1$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = (-t^3+t)\vhat{i} +$$ $$(\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k}$$ at the point $$t=1$$.

$$\langle -\sqrt{2}/2, \sqrt{2}/2, 0 \rangle$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = (-t^3+t)\vhat{i} +$$ $$(\ln(t^2))\vhat{j} + (\cos(\pi t))\vhat{k}$$ at the point $$t=1$$.

Solution

### 2052 video

video by PatrickJMT

$$\langle -\sqrt{2}/2, \sqrt{2}/2, 0 \rangle$$

$$\vec{r}(t) = \langle t\sqrt{2}, e^t, e^{-t} \rangle$$, $$t=0$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle t\sqrt{2}, e^t, e^{-t} \rangle$$ at the point $$t=0$$.

$$\vec{T}(0) = \langle \sqrt{2}/2, 1/2, -1/2 \rangle$$

Problem Statement

Calculate the unit tangent vector for the vector function $$\vec{r}(t) = \langle t\sqrt{2}, e^t, e^{-t} \rangle$$ at the point $$t=0$$.

Solution

### 2053 video

video by David Lippman

$$\vec{T}(0) = \langle \sqrt{2}/2, 1/2, -1/2 \rangle$$