Kissing Curves

At exactly what values of a and x do the curves y = ax and y = loga(x) kiss?

Drag the slider value for a to explore this problem using the graph, then try to answer the questions that follow.

Note: use e for exp(1) in your responses to the questions below.

 

Before beginning the assessment task, you should activate the GeoGebra CAS engine - press the Activate CAS button which follows until you get a correct result. It may take a couple of attempts. Then commence the task.

 

  

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Assessment

Be sure you have activated CAS before you begin.

Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?

 

1. Drag the slider above to find the value of a when the two curves kiss.

 

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2. Study the graphs above and see if you can determine the value for x where the two curves kiss exactly. x =

 

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3. Knowing the value for x where the two curves meet, write an equation to solve for a to find where the curves kiss exactly? Solve...

 

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4. The exact solution for a =

 

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How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

 

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Steve Arnold 2018, Created with GeoGebra

This dynamic figure was created using the latest version of the free GXWeb from Saltire Software, as shown below.

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©2019 Compass Learning Technologies ← Live Mathematics on the Web ← GeoGebra Showcase ← Kissing Curves