Intermediate Geometry : How to find the slope of a tangent line

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #2 : Tangent Lines

What is the slope of the tangent line to the graph of \(\displaystyle y=x^2-6\) when \(\displaystyle x=3\)?

Possible Answers:

\(\displaystyle 9\)

\(\displaystyle 6\)

\(\displaystyle 3\)

\(\displaystyle 0\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle 6\)

Explanation:

To find the slope of the tangent line, first we must take the derivative of \(\displaystyle x^2-6\), giving us \(\displaystyle 2x\). Next we simply plug in our given x-value, which in this case is \(\displaystyle x=3\). This leaves us with a slope of \(\displaystyle 6\).

Example Question #11 : Tangent Lines

Suppose the equation of a line is \(\displaystyle -3x+5y=-3\).  What is the slope of the tangent line at \(\displaystyle x=3\)?

Possible Answers:

\(\displaystyle -\frac{9}{5}\)

\(\displaystyle \frac{9}{5}\)

\(\displaystyle -3\)

\(\displaystyle \frac{5}{3}\)

\(\displaystyle \frac{3}{5}\)

Correct answer:

\(\displaystyle \frac{3}{5}\)

Explanation:

Rewrite \(\displaystyle -3x+5y=-3\) in slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle 5y=3x-3\)

\(\displaystyle y=\frac{3}{5}x-\frac{3}{5}\)

The slope of the tangent line is \(\displaystyle \frac{3}{5}\).

 

 

Example Question #221 : Coordinate Geometry

Suppose a function \(\displaystyle y=1\).  What is the slope of the tangent line at \(\displaystyle x=10\)?

Possible Answers:

\(\displaystyle 0\)

\(\displaystyle -1\)

\(\displaystyle 10\)

\(\displaystyle 1\)

\(\displaystyle \textup{There is no slope.}\)

Correct answer:

\(\displaystyle 0\)

Explanation:

Write the formula for slope-intercept form.

\(\displaystyle y=mx+b\)

\(\displaystyle m=0\)

The slope of \(\displaystyle y=1\) is always zero at every point on the domain.  Therefore, the slope at \(\displaystyle x=10\) must also be zero.

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