Si(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:

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  .

Use differentials to approximate the value of (33)1

Use a linear approximation (or differentials) to estimate the given number.(1.999)3

Let y = f(x) be the solution to the differential equation 𝑑𝑦𝑑𝑥=𝑓′(𝑥) with initial condition f(2) = 3. Selected values of f' are given in the table above. What is the approximation for f(2.4) if Euler’s method is used, starting at x = 2 with two steps of equal size?

Use a linear approximation (or differentials) to estimate the given number.100.4