Math / Calculus

Question4. Find the linear approximation of the following functions $f$ at the given $x$ -value and use it to approximate the function at the given value. a.) $f(\theta)=\sin \theta, \quad \theta=0, \sin (0.2) \approx$ ? b.) $f(x)=\ln (1+2 x), x=0, \ln (1.2) \approx$ ?

Studdy Solution

STEP 1

1. We are using linear approximation (or linearization) to estimate the value of a function near a given point.
2. Linear approximation uses the formula $L(x) = f(a) + f'(a)(x-a)$ , where $a$ is the point of approximation.
3. We are given specific functions and points to approximate.

STEP 2

1. Linear approximation for $f(\theta) = \sin \theta$ at $\theta = 0$ .
2. Use the linear approximation to estimate $\sin(0.2)$ .
3. Linear approximation for $f(x) = \ln(1+2x)$ at $x = 0$ .
4. Use the linear approximation to estimate $\ln(1.2)$ .

STEP 3

For $f(\theta) = \sin \theta$ , calculate $f(0)$ and $f'(0)$ .
- $f(0) = \sin(0) = 0$ - $f'(\theta) = \cos \theta$ , so $f'(0) = \cos(0) = 1$

STEP 4

Use the linear approximation formula $L(\theta) = f(0) + f'(0)(\theta - 0)$ .
- $L(\theta) = 0 + 1 \cdot (\theta - 0) = \theta$
Approximate $\sin(0.2)$ using $L(\theta)$ :
- $L(0.2) = 0.2$

STEP 5

For $f(x) = \ln(1+2x)$ , calculate $f(0)$ and $f'(0)$ .
- $f(0) = \ln(1+2 \cdot 0) = \ln(1) = 0$ - $f'(x) = \frac{2}{1+2x}$ , so $f'(0) = \frac{2}{1+2 \cdot 0} = 2$

STEP 6

Use the linear approximation formula $L(x) = f(0) + f'(0)(x - 0)$ .
- $L(x) = 0 + 2 \cdot (x - 0) = 2x$
Approximate $\ln(1.2)$ using $L(x)$ :
- Since $\ln(1.2) = \ln(1 + 2 \cdot 0.1)$ , use $L(0.1) = 2 \cdot 0.1 = 0.2$
The approximations are:
a.) $\sin(0.2) \approx 0.2$
b.) $\ln(1.2) \approx 0.2$

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