Difference between revisions of "009C Sample Final 3, Problem 10"
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!Step 2: | !Step 2: | ||
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| − | |Now, the origin corresponds to <math>x=0</math> and <math>y=0.</math> | + | |Now, the origin corresponds to <math style="vertical-align: 0px">x=0</math> and <math style="vertical-align: -4px">y=0.</math> |
|- | |- | ||
| − | |This gives us two equations. When we solve for <math>t,</math> we get <math>t=0.</math> | + | |This gives us two equations. When we solve for <math style="vertical-align: -4px">t,</math> we get <math style="vertical-align: -1px">t=0.</math> |
|- | |- | ||
| − | |Plugging in <math>t=0</math> into | + | |Plugging in <math style="vertical-align: -1px">t=0</math> into |
|- | |- | ||
| <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t},</math> | | <math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t},</math> | ||
|- | |- | ||
| − | |we see that <math>\frac{dy}{dx}</math> is undefined at <math>t=0.</math> | + | |we see that <math style="vertical-align: -14px">\frac{dy}{dx}</math> is undefined at <math style="vertical-align: -1px">t=0.</math> |
|- | |- | ||
|So, there is no tangent line at the origin. | |So, there is no tangent line at the origin. | ||
Revision as of 13:50, 5 March 2017
A curve is described parametrically by
(a) Sketch the curve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2\le t \le 2.}
(b) Find the equation of the tangent line to the curve at the origin.
| Foundations: |
|---|
| 1. What two pieces of information do you need to write the equation of a line? |
|
You need the slope of the line and a point on the line. |
| 2. What is the slope of the tangent line of a parametric curve? |
|
The slope is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.} |
Solution:
| (a) |
|---|
| Insert graph |
(b)
| Step 1: |
|---|
| First, we need to find the slope of the tangent line. |
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dt}=3t^2-1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt}=2t,} we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t}.} |
| Step 2: |
|---|
| Now, the origin corresponds to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0.} |
| This gives us two equations. When we solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t,} we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0.} |
| Plugging in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0} into |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{3t^2-1}{2t},} |
| we see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}} is undefined at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0.} |
| So, there is no tangent line at the origin. |
| Final Answer: |
|---|
| (a) See above |
| (b) There is no tangent line at the origin. |