You CAN Ace Calculus

 vectors vector functions

### 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

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.

For the following discussion, we will consider a parameterized curve defined by the vector function $$\displaystyle{ \vec{r}(t) = \langle x(t), y(t), z(t) \rangle }$$ which is traversed once on the continuous interval $$a \leq t \leq b$$.
Some books use the Greek letter $$\kappa$$ (kappa) for curvature. We use a capital K.

The curvature of a smooth curve is a measure of how 'tight' or 'sharp' the curve is. If we have a smooth curve $$\vec{r}$$ and we have a function s which is the arc length function, the curvature is defined to be
$$\displaystyle{ K(s) = \left\| \frac{d\vec{T}}{ds} \right\|}$$.

This equation for the curvature is not particularly useful for calculations. So we have several other ways to write the equation of the curvature. But first notice, that the curvature is a scalar function, not a vector function. And since it is the norm of a vector, the curvature will always be positive.

Curvature Formula #1

For our first equation to use when calculating the curvature, we will use the chain rule to write $$\displaystyle{ \frac{d\vec{T}}{dt} = \frac{d\vec{T}}{ds} \cdot \frac{ds}{dt} }$$. We can solve for $$d\vec{T}/ds$$ to get

$$\displaystyle{ K = \left\| \frac{d\vec{T}}{ds} \right\| = \frac{\| d\vec{T}/dt \|}{ \| ds/dt \| } = \frac{1}{\| \vec{v} \|} \left\| \frac{d\vec{T}}{dt} \right\| }$$

Notice in the previous equation, we used $$ds/dt = \|\vec{v}\|$$ to simplify the equation somewhat.
Now we can write the first curvature formula in a form that we can use for calculations.

$$\displaystyle{ K(t) = \frac{1}{\|\vec{v}\|} \left\| \frac{d\vec{T}}{dt} \right\| = \frac{\| \vec{T}'(t) \|}{\|\vec{r}'(t)\|} }$$

Curvature Formula #2

If we define a vector $$\vec{a} = d\vec{v}/dt$$ as an acceleration vector, a second curvature formula is
$$\displaystyle{ K=\frac{\|\vec{v} \times \vec{a} \|}{\|\vec{v}\|^3} }$$

Before working some practice problems, here is a quick video clip for you that should help you understand curvature a bit better.

video by MIP4U

### 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 curvature of these vector functions. Give your answers in exact, completely factored form.

$$\vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} +$$ $$4\cos t\vhat{k}$$

Problem Statement

$$\vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} +$$ $$4\cos t\vhat{k}$$

$$K(t) = 4/25$$

Problem Statement

$$\vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} +$$ $$4\cos t\vhat{k}$$

Solution

### 2073 solution video

video by Krista King Math

$$K(t) = 4/25$$

$$\vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle$$

Problem Statement

$$\vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle$$

$$K(t) = 1/2$$

Problem Statement

$$\vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle$$

Solution

### 2074 solution video

video by MIP4U

$$K(t) = 1/2$$

Find the curvature at $$t=1$$ for $$\vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle$$.

Problem Statement

Find the curvature at $$t=1$$ for $$\vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle$$.

$$K(1) = \sqrt{2}/3$$

Problem Statement

Find the curvature at $$t=1$$ for $$\vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle$$.

Solution

### 2075 solution video

video by MIP4U

$$K(1) = \sqrt{2}/3$$

Find the curvature of $$\vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle$$ at $$t = 2$$.

Problem Statement

Find the curvature of $$\vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle$$ at $$t = 2$$.

$$K(t) = 24\sqrt{3229}/(9817)^{3/2}$$

Problem Statement

Find the curvature of $$\vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle$$ at $$t = 2$$.

Solution

### 2076 solution video

video by MIP4U

$$K(t) = 24\sqrt{3229}/(9817)^{3/2}$$

Find the curvature of $$\vec{r}(t) = \langle 2\cos(3t),2\sin(3t),$$ $$4t \rangle$$ at $$t = 0$$.

Problem Statement

Find the curvature of $$\vec{r}(t) = \langle 2\cos(3t),2\sin(3t),$$ $$4t \rangle$$ at $$t = 0$$.

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Problem Statement

Find the curvature of $$\vec{r}(t) = \langle 2\cos(3t),2\sin(3t),$$ $$4t \rangle$$ at $$t = 0$$.

Solution

This problem is solved in each of these videos. The first video uses the cross product version of the curvature formula, the second video does not use the cross product.

video by MIP4U

video by MIP4U