Evaluating integrals can be difficult. So let's instead use the total differential approximation to approximate the Sine IntegralSi(x)=∫0xsin(t)tdtSi⁡(𝑥)=∫0𝑥sin⁡(𝑡)𝑡𝑑𝑡 near the point A=[3,Si(3)]≈[3,1.849]𝐴=[3,Si⁡(3)]≈[3,1.849] . The differential approximation to  SiSi near 33 is given by the formulaSi(x)≈1.849+Si′(3)(x−3)Si⁡(𝑥)≈1.849+Si′⁡(3)(𝑥−3) .  Since  Si′(3)=Si′⁡(3)=     , we have the linear approximation to Si(x)Si⁡(𝑥)   near x=3𝑥=3 : Si(x)≈Si⁡(𝑥)≈   . Use the linear approximation to evaluate (to 3 decimal places) Si(2.9)≈Si⁡(2.9)≈ IncorrectYour Answer: 1.84  .

Para aproximar la Integral Seno $Si(x) = \int_0^x \frac{\sin(t)}{t} dt$ cerca del punto $A = [3, Si(3)] \approx [3, 1.849]$, utilizamos la fórmula de la aproximación diferencial:

$Si(x) \approx 1.849 + Si'(3)(x - 3)$

Primero, necesitamos calcular $Si'(x)$. Sabemos que:

$Si'(x) = \frac{\sin(x)}{x}$

Entonces, evaluamos $Si'(3)$:

$Si'(3) = \frac{\sin(3)}{3}$

Calculamos el valor numérico de $\sin(3)$:

$\sin(3) \approx 0.1411$

Por lo tanto:

$Si'(3) \approx \frac{0.1411}{3} \approx 0.047$

Ahora, utilizamos esta derivada para la aproximación lineal:

$Si(x) \approx 1.849 + 0.047(x - 3)$

Para $x = 2.9$:

$Si(2.9) \approx 1.849 + 0.047(2.9 - 3)$

$Si(2.9) \approx 1.849 + 0.047(-0.1)$

$Si(2.9) \approx 1.849 - 0.0047$

$Si(2.9) \approx 1.8443$

Redondeando a tres decimales:

$Si(2.9) \approx 1.844$

Por lo tanto, la aproximación lineal de $Si(2.9)$ es aproximadamente 1.844.