**High School Number Sense Lesson 75: Finding the Slope of a Line Tangent to a Curve**

Last month we introduced slopes in the

**SlopeLineEqua**post. While that lesson dealt with linear equations, today's lesson covers curves. It is an extension of Saturday's lesson on taking the derivative of a function (

**FunctionDeriv**). This concept appeared

**3 times**last year, with a median placement at

**question # 76**.

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**Definition:**

The

**tangent**line to a curve at a given point is the straight line that "just touches" the curve at that point. It is the best straight-line approximation to the curve at that point. It is also the

**slope**of the curve at that point. Let's take a look at how to find this slope...which will look very familiar if you've studied last Saturday's post.

**How to Solve:**

- Take the
**derivative**of the function. - Substitute the
**x value**of the line (or point) into the result of Step 1.

**Example 1:**

**The slope of the line tangent to f(x) = x**

^{2}- 5x + 4 at (-1, 10) is ___1. Take the derivative:

**f'(x) = 2x - 5**.

2. Substitute

**-1**for x: f'(-1) = 2(-1) - 5 = -2 - 5 =

**-7**.

**Example 2:**

**The slope of the line tangent to f(x) = x**

^{2}+ 2x + 1 at (2, 9) is ___1. Take the derivative:

**f'(x) = 2x + 2**.

2. Substitute

**2**for x: f'(2) = 2(2) + 2 = 4 + 2 =

**6**.

**Example 3:**

**Find the slope of the line tangent to y = 3x**.

^{2}+ 2x - 8 at x = -21. Take the derivative:

**y' = 6x + 2**.

2. Substitute

**-2**for x: y' = 6(-2) + 2 = -12 + 2 =

**-10**.

**Example 4:**

**The slope of the line tangent to y = 5x**

^{2}- 3x + 1 at the point (2, 15) is ___1. Take the derivative:

**y' = 10x - 3**.

2. Substitute

**2**for x: y' = 10(2) - 3 = 20 - 3 =

**17**.

**Example 5:**

**The slope of the line tangent to y = x**.

^{3}- x + 1 at the point (-1, 1) is ___1. Take the derivative:

**y' = 3x**.

^{2}- 12. Substitute

**-1**for x: y' = 3(-1)

^{2}- 1 = 3(1) - 1 = 3 - 1 =

**2**.

**Check back soon for a free worksheet to help you practice SlopeLineTang.**

**Up Next for High School: AddF1/n(n+1)**