#Local Linearity of a Function

#Table of Contents

1. What Does Local Linearity Mean?
2. Using a Tangent to Approximate a Function
3. Overestimate or Underestimate?
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

#What Does Local Linearity Mean?

For instance, consider the function $y = x^3 - 4x + 3$. At the point (2, 3), the tangent line $y = 8x - 13$ approximates the curve closely around this point. As you zoom in around (2, 3), the curve looks more like the tangent line.

#Using a Tangent to Approximate a Function

#Equation of the Tangent

The equation of the tangent to $f(x)$ at $x = a$ is given by:

$$y - f(a) = f'(a)(x - a)$$

or

$$y = f(a) + f'(a)(x - a)$$

Provided that $f(x)$ is differentiable at $a$.

Due to the local linearity of a function, this can be a linear approximation for $f(x)$ at points close to $(a, f(a))$.

For the graph of $y = x^3 - 4x + 3$ at (2, 3), the tangent is $y = 8x - 13$. The tangent will be an approximation for the curve close to (2, 3).

#Approximated Values vs Real Values

The table shows how the approximation is more accurate closer to the point where the tangent intersects the curve. It will be less accurate further away from the point of intersection.

Using a tangent to approximate a curve within a small interval can simplify calculations or computational processes, although this comes with a trade-off in accuracy.

#Overestimate or Underestimate?

The values of the function approximated by the tangent will either be an overestimate or underestimate of the real value. This depends on the concavity of the function at the point where the tangent intersects the curve.

In the graph of $f(x)$, if the tangent at $x = -2$ will give an overestimate as the function is concave down at this point, and the tangent at $x = 2$ will give an underestimate as the function is concave up at this point.

#Worked Example

#(a) Find an approximation to the value of $\sqrt{65}$ using a linear approximation of $y = \sqrt{x}$.

Answer: We can use a linear approximation by finding a tangent to the graph of $y = \sqrt{x}$ at a point we know the coordinates of, and which is close to $x = 65$.

Use the point (64, 8) as this is close to 65 and has integer coordinates to make working easier.

Tangent to $f(x) = \sqrt{x} = x^{1/2}$ at (64, 8):

For the equation of the tangent, use the form $y - y_1 = m(x - x_1)$:

$$y - f(64) = f'(64)(x - 64)$$

Find $f'(64)$:

$$f'(x) = \frac{1}{2}x^{-1/2} \quad \text{and} \quad f'(64) = \frac{1}{2} \cdot \frac{1}{\sqrt{64}} = \frac{1}{16}$$

Find the equation of the tangent:

$$y - 8 = \frac{1}{16}(x - 64)$$

Use the tangent to estimate the value of $\sqrt{65}$ by substituting $x = 65$:

$$y - 8 = \frac{1}{16}(65 - 64) \implies y = 8 + \frac{1}{16} = 8.0625$$

An approximation of $\sqrt{65}$ is 8.0625. ### (b) Without calculating the real value of $\sqrt{65}$, explain whether your approximation will be an overestimate or underestimate.

Answer: Consider the concavity of $f(x) = \sqrt{x}$ to decide if the tangent at $x = 64$ will be an over- or underestimate.

$$f'(x) = \frac{1}{2}x^{-1/2} \quad \text{and} \quad f''(x) = -\frac{1}{4}x^{-3/2}$$

Since $x$ is always positive, the second derivative is always negative, so the graph of $y = \sqrt{x}$ is always concave down. Therefore, the approximation using a tangent will be an overestimate.

You can also see this when sketching a graph of $y = \sqrt{x}$ and the tangent at (64, 8). The tangent is always above the curve, so it will be an overestimate.

#Practice Questions

Practice Question

1. Find the linear approximation of $f(x) = \ln(x)$ at $x = 1$ and use it to estimate $\ln(1.05)$.

Practice Question

2. Determine if the linear approximation of $f(x) = e^x$ at $x = 0$ will be an overestimate or underestimate for $x = 0.1$.

Practice Question

3. Use the tangent line approximation to find an estimate for $\cos(0.1)$ using $f(x) = \cos(x)$ at $x = 0$.

#Glossary

• Local Linearity: The property of a function whereby its graph looks like a straight line when zoomed in sufficiently close to a point.
• Tangent: A straight line that touches a curve at a single point without crossing it.
• Concavity: The attribute of a curve that describes whether it bends upwards or downwards.
• Overestimate: An approximation that is higher than the actual value.
• Underestimate: An approximation that is lower than the actual value.

#Summary and Key Takeaways

• Local Linearity: Functions appear linear when zoomed in sufficiently close to a point.
• Tangent Lines: Used as linear approximations for functions at a point.
• Accuracy: The closer to the point of tangency, the more accurate the approximation.
• Concavity and Estimates: Determines whether the linear approximation is an overestimate or underestimate.