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AP Calculus BC
Unit 4: Contextual Applications of Differentiation
4.6 Approximating Values of a Function Using Local Linearity
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What is the main concept behind local linearity?
Curve near point resembles straight line
Local linearity allows us to approximate function values using the
tangent
line.
Local linearity implies that a curve behaves like a
straight line
near a specific point.
The tangent line to
f
(
x
)
=
f(x) =
f
(
x
)
=
x
+
x +
x
+
2
2
2
at
x
=
x =
x
=
3
3
3
is f(x) = x + 2</latex> because the function is already a straight line.
What is the approximation of
f
(
3.01
)
f(3.01)
f
(
3.01
)
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
+
x +
x
+
2
2
2
using the tangent line at
x
=
x =
x
=
3
3
3
?
5.01
The tangent line at a point on a
curve
provides an accurate approximation of the function's value nearby.
What two components are needed to find the equation of a tangent line to a function at a specific point?
Slope and a point
Steps to find the equation of the tangent line to f(x) = x^2 - 3x + 2</latex> at
x
=
x =
x
=
2
2
2
1️⃣ Find the derivative:
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
2
x
−
3
2x - 3
2
x
−
3
2️⃣ Evaluate the derivative at
x
=
x =
x
=
2
2
2
:
f
′
(
2
)
=
f'(2) =
f
′
(
2
)
=
1
1
1
3️⃣ Find the y-coordinate at
x
=
x =
x
=
2
2
2
:
f
(
2
)
=
f(2) =
f
(
2
)
=
0
0
0
4️⃣ Use the point-slope form:
y
=
y =
y
=
x
−
2
x - 2
x
−
2
The point-slope form of a line is
y
−
y
1
=
y - y_{1} =
y
−
y
1
=
m
(
x
−
x
1
)
m(x - x_{1})
m
(
x
−
x
1
)
, where
m
m
m
represents the slope of the line.
What is the slope of the tangent line to
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
at
x
=
x =
x
=
2
2
2
?
1
The tangent line approximation method assumes that the curve behaves like a straight line near the
point of tangency
.
Steps to approximate
f
(
2.01
)
f(2.01)
f
(
2.01
)
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
using the tangent line at
x
=
x =
x
=
2
2
2
1️⃣ Find the derivative:
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
2
x
−
3
2x - 3
2
x
−
3
2️⃣ Evaluate the derivative at
x
=
x =
x
=
2
2
2
:
f
′
(
2
)
=
f'(2) =
f
′
(
2
)
=
1
1
1
3️⃣ Calculate the point
(
2
,
f
(
2
)
)
(2, f(2))
(
2
,
f
(
2
))
:
f
(
2
)
=
f(2) =
f
(
2
)
=
0
0
0
4️⃣ Write the equation of the tangent line:
y
=
y =
y
=
x
−
2
x - 2
x
−
2
5️⃣ Approximate
f
(
2.01
)
f(2.01)
f
(
2.01
)
:
f
(
2.01
)
≈
0.01
f(2.01) ≈ 0.01
f
(
2.01
)
≈
0.01
What is the actual value of
f
(
2.01
)
f(2.01)
f
(
2.01
)
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
?
0.0301
The formula for local linearity is f(x) ≈ f(a) + f'(a)(x - a)</latex>, where
f
(
a
)
f(a)
f
(
a
)
is the value of the function at a.
Local linearity assumes that the curve near a point resembles a
straight line
.
What is the local linearity approximation of
f
(
3.01
)
f(3.01)
f
(
3.01
)
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
+
x +
x
+
2
2
2
at
x
=
x =
x
=
3
3
3
?
5.01
Match the term with its definition:
Local Linearity ↔️ Curve near a point resembles a straight line
Tangent Line ↔️ Line that touches the curve at one point
Approximation ↔️ Estimate of a function's value
What is the formula for local linearity?
f
(
x
)
≈
f
(
a
)
+
f(x) ≈ f(a) +
f
(
x
)
≈
f
(
a
)
+
f
′
(
a
)
(
x
−
a
)
f'(a)(x - a)
f
′
(
a
)
(
x
−
a
)
In the local linearity formula,
f
(
a
)
f(a)
f
(
a
)
represents the value of the function at point a
What does
f
′
(
a
)
f'(a)
f
′
(
a
)
represent in the local linearity formula?
Derivative at
x
=
x =
x
=
a
a
a
For the function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
+
x +
x
+
2
2
2
at
x
=
x =
x
=
3
3
3
, the tangent line approximation of
f
(
3.01
)
f(3.01)
f
(
3.01
)
is 5.01
The tangent line approximation of
f
(
3.01
)
f(3.01)
f
(
3.01
)
for f(x) = x + 2</latex> at
x
=
x =
x
=
3
3
3
is
5.01
5.01
5.01
.
The tangent line approximation for
f
(
3.01
)
f(3.01)
f
(
3.01
)
of the function f(x) = x + 2</latex> is exact.
Steps to find the equation of the tangent line to a function
f
(
x
)
f(x)
f
(
x
)
at
x
=
x =
x
=
a
a
a
1️⃣ Find the derivative
f
′
(
x
)
f'(x)
f
′
(
x
)
2️⃣ Evaluate
f
′
(
a
)
f'(a)
f
′
(
a
)
to get the slope
m
m
m
3️⃣ Find the corresponding y-coordinate
f
(
a
)
f(a)
f
(
a
)
4️⃣ Use the point-slope form:
y
−
f
(
a
)
=
y - f(a) =
y
−
f
(
a
)
=
m
(
x
−
a
)
m(x - a)
m
(
x
−
a
)
What does local linearity describe about a curve near a point?
Resembles a straight line
What does
f
′
(
a
)
f'(a)
f
′
(
a
)
represent in the equation of the tangent line?
Slope of the tangent line
The tangent line to
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
at
x
=
x =
x
=
2
2
2
has the equation
y
=
y =
y
=
x
−
2
x - 2
x
−
2
.
The formula for the tangent line is f(x) ≈
f
(
a
)
+
f(a) +
f
(
a
)
+
f
′
(
a
)
(
x
−
a
)
f'(a)(x - a)
f
′
(
a
)
(
x
−
a
)
The value of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
at
x
=
x =
x
=
2
2
2
is 0
What is the first step in using the tangent line approximation method?
Find the derivative
In the tangent line approximation,
f
′
(
a
)
f'(a)
f
′
(
a
)
is used to calculate the slope
What does
f
(
a
)
f(a)
f
(
a
)
represent in the tangent line formula?
Function value at
x
=
x =
x
=
a
a
a
The tangent line approximation always gives the exact value of the function.
False
The tangent line approximation of
f
(
3.01
)
f(3.01)
f
(
3.01
)
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
+
x +
x
+
2
2
2
is
5.01
5.01
5.01
Match the term with its definition:
Error ↔️ Difference between actual and approximated values
Tangent Line Approximation ↔️ Method to estimate function values
Local Linearity ↔️ Curve resembles a straight line
Steps to find the equation of the tangent line to a function
f
(
x
)
f(x)
f
(
x
)
at a point
x
=
x =
x
=
a
a
a
1️⃣ Find the derivative
f
′
(
x
)
f'(x)
f
′
(
x
)
2️⃣ Evaluate the derivative at
x
=
x =
x
=
a
a
a
to get the slope
m
=
m =
m
=
f
′
(
a
)
f'(a)
f
′
(
a
)
3️⃣ Find the corresponding y-coordinate
f
(
a
)
f(a)
f
(
a
)
4️⃣ Use the point-slope form to write the equation:
y
−
f
(
a
)
=
y - f(a) =
y
−
f
(
a
)
=
m
(
x
−
a
)
m(x - a)
m
(
x
−
a
)
The derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
at
x
=
x =
x
=
2
2
2
is 1
What is the formula for calculating the error in local linear approximation?
E
r
r
o
r
=
Error =
E
rror
=
∣
f
(
x
)
−
L
(
x
)
∣
|f(x) - L(x)|
∣
f
(
x
)
−
L
(
x
)
∣
Local linearity occurs when a curve near a point closely resembles a
line
What is the equation of the tangent line to
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
3
x
+
x^{2} - 3x +
x
2
−
3
x
+
2
2
2
at
x
=
x =
x
=
2
2
2
?
y
=
y =
y
=
x
−
2
x - 2
x
−
2
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