4.6 Approximating Values of a Function Using Local Linearity

• The tangent line to $f(x) =$$x^{2}$ at x = 2</latex> is $y =$$4x - 4$, which can be used to approximate f(2.1).
• What is the formula for approximating $f(x + \Delta x)$ using the tangent line approximation?


  f(x) + f'(x) \Delta x</latex>
• The tangent line approximation is effective for estimating function values near a known point.


  True
• Why is the tangent line approximation considered simple and effective?

  Estimates values near a point
• When approximating $\sqrt{4.1}$ using the tangent line to $f(x) =$$\sqrt{x}$ at $x =$$4$, the value of $\Delta x$ is 0.1.
• The tangent line approximation formula is $f(x + \Delta x) \approx f(x) +$$f'(x) \Delta x$


  True
• In the example, $\Delta x$ is equal to 0.1
• Match the component of the tangent line approximation with its description:

  $f(x)$ ↔️ Function value at $x$

  $f'(x)$ ↔️ Derivative of $f$ at $x$

  $\Delta x$ ↔️ Change in $x$

  $f(x + \Delta x)$ ↔️ Approximate function value at $x +$$\Delta x$
• The derivative of $f(x) =$$\sqrt{x}$ is $f'(x) =$$\frac{1}{2\sqrt{x}}$


  True
• What is the definition of local linearity?

  Approximation by a tangent line
• What is the tangent line to f(x) = x^{2}</latex> at $x =$$2$?


  $y =$$4x - 4$
• What is the principle underlying the tangent line approximation method?

  Local linearity
• The tangent line approximation estimates the function value at $x$.


  False
• What is the accuracy of the tangent line approximation dependent on?

  Small $\Delta x$
• To approximate $\sqrt{4.1}$ using the tangent line to $f(x) =$$\sqrt{x}$ at $x =$$4$, $\Delta x$ is equal to 0.1
• What is the formula for differential approximation?

  $df \approx f'(x) dx$
• Differentials provide a more direct estimate of the change in $f(x)$ rather than estimating the absolute value
• Differentials are more useful for estimating absolute values than changes in function values.

  False
• The tangent line approximation formula is f(x + \Delta x) \approx f(x) + f'(x) \Delta x</latex>, which estimates the absolute value
• The differential approximation estimates the change in f(x)</latex> from $f(9)$ to $f(9.1)$ as $0.0167$.


  True
• The tangent line approximation relies on the principle of local linearity.
• What does $\Delta x$ represent in the tangent line approximation formula?


  Change in $x$
• What is the value of $f(4)$ for $f(x) =$$\sqrt{x}$?


  2
• The tangent line approximation is most accurate near the point of tangency.


  True
• The tangent line approximation relies on the principle of local linearity
• The tangent line approximation is a precise method for estimating function values.

  False
• The tangent line approximation relies on the principle of local linearity.


  True
• $f(x)$ in the tangent line approximation represents the function value at the point $x$


  True
• $\Delta x$ in the tangent line approximation is the change in $x$ from the known point


  True
• The tangent line approximation relies on the principle of local linearity


  True
• Match the feature with its description:

  Purpose of differentials ↔️ Estimate change in $df$

  Purpose of tangent line approximation ↔️ Estimate $f(x + dx)$
• What is the purpose of differentials in local linearity?

  Estimate change in function value
• What is the approximate change in $f(x) =$$x^{2}$ when $x$ changes from $2$ to $2.1$?


  $0.4$
• What does the differential approximation estimate?

  Change in function value
• The tangent line approximation estimates the function value at a new point
• For $f(x) =$$\sqrt{x}$, the tangent line approximation at $x =$$9$ with $\Delta x =$$0.1$ gives $f(9.1) \approx 3.0167$.


  True
• Local linearity provides high accuracy for approximations near the point of tangency.


  True
• The tangent line approximation relies on the principle of local linearity to estimate function values near a known point.
• To approximate $\sqrt{4.1}$ using the tangent line to $f(x) =$$\sqrt{x}$ at $x =$$4$, the approximation is 2.025.
• When approximating $\sqrt{4.1}$ using local linearity, the value $f'(4)$ is \frac{1}{4}.