4.6 Approximating Values of a Function Using Local Linearity

• What is the concept of local linearity?

  Approximating a function with its tangent line
• Consider the function f(x) = x² at x = 2. The value of its derivative at x = 2 is 4
• The absolute error in local linearity increases as we move further from the point of tangency
• What is the general equation of the tangent line to a function at a point a?


  y = f(a) + f'(a)(x - a)
• Steps to identify the tangent line on a function's graph:

  1️⃣ Find the derivative of the function

  2️⃣ Locate the point of tangency

  3️⃣ Draw the tangent line
• Steps to identify a tangent line on a function's graph:

  1️⃣ Find the derivative of the function

  2️⃣ Locate the point of tangency

  3️⃣ Draw the tangent line using the point of tangency and the slope from the derivative
• The tangent line touches the function at exactly one point.


  True
• The equation of the tangent line to f(x) = x² at x = 2 is y = 4 + 4(x - 2)
• Local linearity approximates a function near a point with its tangent line.
• What is the limitation of local linearity?

  The error increases further away
• To identify a tangent line, you must first find the derivative of the function.
• What is the formula for the equation of the tangent line to a function f(x) at the point (a, f(a))?

  $y =$$f(a) +$$f'(a)(x - a)$
• In the tangent line equation, f(a) represents the value of the function at the point of tangency
• What are two properties ensured by the tangent line formula for a function f(x) at a point (a, f(a))?

  Passes through (a, f(a)) and has slope f'(a)
• The tangent line to f(x) = x² at x = 2 has a slope of 4.


  True
• What is the concept of local linearity?

  Approximating a function with its tangent line
• A tangent line intersects a function's graph at a single point
• What is the equation of the tangent line to the function f(x) at x = 2?

  y = 4 + 4(x - 2)
• Steps to identify a tangent line on a function's graph

  1️⃣ Find the derivative of the function

  2️⃣ Locate the point of tangency

  3️⃣ Draw the tangent line using the point and slope
• What is the derivative of f(x) = x²?

  f'(x) = 2x
• The tangent line to f(x) = x² at x = 2 passes through the point (2, 4).


  True
• The equation of the tangent line to f(x) = x² at x = 2 is y = 4 + 4(x - 2
• f(a) in the tangent line equation represents the value of the function at the point of tangency.


  True
• How is the tangent line used to approximate function values near a point of tangency?

  Local linear approximation
• Using the tangent line approximation, the value of f(2.1) for f(x) = x² is approximately 4.4
• Greater curvature of the function increases the error in local linearity.


  True
• The tangent line equation is used to approximate the value of f(2.1)
• What is the actual value of f(2.1)?

  4.41
• Greater curvature of the function leads to higher errors in local linearity.


  True
• For $f(x) =$$x^{2}$ and $a =$$2$, what is the tangent line equation used to approximate $f(2.1)$?


  $L(x) =$$4 +$$4(x - 2)$
• Local linearity is a good approximation when the function is continuous and differentiable, and the region considered is small.


  True
• What is the absolute error between the actual value of f(x) = x² and its tangent line approximation at x = 1.8?


  0.04
• Match the properties of a tangent line with their descriptions:

  Intersects the function at a single point ↔️ Touches the function at the point of tangency

  Same slope as the function at that point ↔️ Equal to the derivative at the point of tangency
• At x = 2, the slope of the function f(x) = x² and its tangent line are both equal to 4
• What is the equation of the tangent line to f(x) = x² at x = 2?


  y = 4 + 4(x - 2)
• The derivative of a function gives the slope of the tangent line at any point.
• What is the derivative of f(x) = x²?

  f'(x) = 2x
• The absolute error between the tangent line approximation and the actual function value increases as you move further from the point of tangency.


  True
• Match the concept with its description:

  Tangent line ↔️ Straight line with the same slope as the function at a single point

  Derivative ↔️ Represents the slope of the function at any point

  Point of tangency ↔️ The specific point where the tangent line touches the function
• The tangent line and the function have the same slope at the point of tangency.


  True