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Wize University Calculus 1 Textbook > Applications of Differentiation

Linearization (Linear Approximation)

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Linearization

Tangent Lines can give us great approximations to functions near a point. The process of using this linear approximation is called Linearization.

Linearization

The linearization or linear approximation, of the functionf(x)f\left(x\right) about x=ax=a is the functionL(x)L\left(x\right) defined by
L(x)=f(a)+f′(a)(x−a)\boxed{L(x)=f(a)+f'(a)(x-a)}
The value of aa is sometimes called the center or the anchor point of the approximation.

Note: The closer the input of L(x)L(x) is to aa , the better the approximation of f(x)f(x) will be.

Over/Under Estimates

If the function is concave up (upward U shaped in the neighborhood of the approximation) the approximation is an underestimate of the actual value.

If the function is concave down (downward U shaped in the neighborhood of the approximation) the approximation is an overestimate of the actual value.

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Example: Linearization

Approximate the square root of 26 without using a calculator.

Take f(x)=x ⇒f′(x)=12x\begin{aligned} f(x)&=\sqrt{x}\\ \text{ }\\ \Rightarrow f'(x)&=\frac{1}{2\sqrt{x}} \end{aligned}



Also, take a=25 ⇒f′(a)=1225=110\begin{aligned} a&=25\\ \text{ }\\ \Rightarrow f'(a)&=\frac{1}{2\sqrt{25}}=\frac{1}{10} \end{aligned}



We have
f(x)≈f(x0)+f′(a)(x−a) ⇒f(26)=26≈f(25)+f′(25)[26−25]=25+110(1)=5+110 ⇒26≈5.1\begin{aligned} f(x) &\approx f(x_0)+f'(a)(x-a)\\ \text{ }\\ \Rightarrow f(26)&=\sqrt{26}\approx f(25)+f'(25)[26-25]\\ &=\sqrt{25}+\frac{1}{10}(1)\\ &=5+\frac{1}{10}\\ \text{ }\\ \Rightarrow \sqrt{26}&\approx 5.1 \end{aligned}

Using the calculator
26≈5.0990195\sqrt{26}\approx 5.0990195
Use a linear approximation to estimate 8.012/38.01^{2/3}
Using a linear approximation, estimate g(1.1)g(1.1), given that g(x)=f(x2+1)g(x)=f(x^2+1) and f(2)=f′(2)=1f(2)=f^{\prime}(2)=1
Extra Practice
Use linearization to construct a linear approximation to x4\sqrt[4]x for xx values near 11, given that f′(x)=14x−34f'(x)=\frac14x^{-\frac34}.
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}. If f(x)f(x) is concave up near this point, what does that tell us about our estimate?
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}. Is your estimate likely to be an over- or under-estimate? HINT: Use f(x)=xf(x)=\sqrt{x}, which gives us f′(x)=12xf'(x)=\frac1{2\sqrt{x}} and f′′(x)=−14x23f''(x)=-\dfrac{1}{4\sqrt[3]{x^2}}.
Linear Approximation of Given Function
Use a linear approximation to estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}.
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2, f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Written Answer 5
Use linear approximation at a suitable close value to estimate arctan⁡(0.9)\arctan(0.9). Your answer may be left in terms of fractions.
Linear Approximation
Approximate (28)1/3(28)^{1/3} using a linear approximation of the function h(x)=x1/3h(x) = x^{1/3}
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value?
The linearization of f(x)=tanxf(x)=tanx at x=π4x=\frac{\pi}{4} is
Linear Approximation
Approximate 93.\sqrt[3]{9}.
Linear Approximation
Use a suitable linear approximation to estimate 1.9951.99^5
Linear Approximation
Use a suitable linear approximation to estimate 3.99\sqrt{3.99}.
Linear Approximation
Use a linear approximation to estimate 9.2\sqrt{9.2} using f(x)=3+x.f(x)=\sqrt{3+x}.
Linear Approximation
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is
Practice: Linear Approximations
Practice: Linear Approximations
Approximate ln⁡(1.01)\ln\left(1.01\right) and ln⁡(0.98)\ln\left(0.98\right).
Find the linear approximation of the following function, and determine if it is an over/under approximation.
f(x)=x34+x3f(x)=\frac{x^3}{4+x^3} at x=2.1x=2.1
final114
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is
Approximate x32+x23 x^{\frac{3}{2}}+x^{\frac{2}{3}} at x=1.1x=1.1 using a linear approximation.
final114
The linear approximation L(x)L(x) of the function f(x)=xf(x)=\sqrt{x} at x=16x=16 is (enter L=...)
Linear Approximation: Estimate using Given Values
Use a linear approximation to estimate f(2.01)f(2.01) given that f(2)=1f(2)=1 and f′(2)=2f^{\prime}(2)=2.
Linear Approximation
Use a suitable linear approximation to estimate 8.01238.01^{\frac{2}{3}}.
Linear Approximation: Estimating a Cube Root
Use a linear approximation to estimate 653 \sqrt[3]{65}. Then use the same linearization equation to estimate 283\sqrt[3]{28}. Is it still a good approximation? Why?
Linear Approximation of Given Function
Use a linear approximation to estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}.
Linear Approximation: Estimating a 4th Root
Use a suitable linear approximation to estimate 68.821/468.82^{1/4} (leave it as an exact value if you're not allowed a calculator on the exam).
Use linearization to construct a linear approximation to x4\sqrt[4]x for xx values near 11.
Use a linear approximation to estimate 99.9\sqrt{99.9}.
Concept Clarifier
Use linear approximation to estimate the value of 603\sqrt[3]{60}.
Linear Approximation
The linear approximation L(x)L(x) of the function f(x)=cos⁡xf(x)=\cos{x} at x=π3x=\frac{\pi}{3} is
Linear Approximation
Using a linear approximation, estimate f(4.1)f(4.1), given that f(4)=2f(4)=2 and f′(x)=x2+20f^{\prime}(x)=\sqrt{x^2+20}. If f(x)f(x) is concave up near this point, what does that tell us about our estimate?
Approximate arctan⁡(0.9)\arctan\left(0.9\right) by linear approximation.
Use a linear approximation to approximate the value of the function by using the given information.
f(2)=1,f′(2)=2f(2)=1, f^{\prime}(2)=2 Approximate f(2.01)f(2.01). Enter your answer to two decimal places.
Use a linear approximation to estimate the value of the following function at the given point.
f(x)=2x+1x3+1\displaystyle f(x)=\frac{2x+1}{x^3+1} Approximate f(1.1)f(1.1).
Use a linear approximation to estimate 821/482^{1/4}.
Approximate x32+x23 x^{\frac{3}{2}}+x^{\frac{2}{3}} at x=1.1x=1.1 using linear approximation about x0=1x_0=1.
Use a linear approximation to estimate the given quantity.
8.012/38.01^{2/3}
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Linear Approximation
Consider the function
f(x)=4−xf(x) = \sqrt{ 4 - x}
Practice: Relative Error
A farmer used 100m of fence material to create a square pen to hold in his animals. However, there's a possible error of 2m in the length of the fence he used.
a) Use a linear approximation to estimate the maximum possible absolute error in the area of the pen. Include units in the answer.
b) Use the previous linear approximation to find the maximum relative error in the area. Include units in your answer.
Practice: Linear Approximation
Consider the function
f(x)=sin⁡(x)f(x) = \sin(x)
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Linear Approximation and Trigonometric Derivative
a) Show that for f(x)=arcsin⁡(x)f(x) = \arcsin(x) , that f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}} .
b) Use linear approximation at a suitable close value to estimate arcsin⁡(0.1)\arcsin(0.1) . Your solution may be left in terms of fractions.
Consider the function
f(x)=4−xf(x) = \sqrt{ 4 - x}
a) Find the linear approximation of the function f(x)f(x) near x=0x = 0 .
Using a linear approximation, estimate g(1.1)g(1.1), given that g(x)=f(x2+1)g(x)=f(x^2+1) and f(2)=f′(2)=1f(2)=f^{\prime}(2)=1.
Use a linear approximation to estimate
8.012/38.01^{2/3}
For the following exercises, use a linear approximation to estimate the given quantities.
8.012/38.01^{2/3}