Using the first order Taylor approximation at the point (﻿x equals﻿0.5, ﻿y equals﻿0.6) for ﻿f left parenthesis x comma y right parenthesis equals sin left parenthesis straight pi y x squared right parenthesis﻿, estimate ﻿f left parenthesis﻿0.7﻿comma﻿0.9﻿right parenthesis﻿.

Give your answer using 2 decimal places and use radians in your calculations.

To estimate \( f(0.7, 0.9) \) using the first-order Taylor approximation at the point \( (x_0, y_0) = (0.5, 0.6) \) for the function \( f(x, y) = \sin(\pi y x^2) \), we follow these steps:

### 1. **Compute \( f(x_0, y_0) \):**
\[
f(0.5, 0.6) = \sin(\pi \times 0.6 \times (0.5)^2) = \sin(0.75 \times \pi \times 0.25) = \sin\left(\frac{3\pi}{16}\right)
\]
\[
\sin\left(\frac{3\pi}{16}\right) \approx 0.55557
\]

### 2. **Compute the partial derivatives:**

**Partial derivative with respect to \( x \):**
\[
f_x(x, y) = \frac{\partial}{\partial x} \sin(\pi y x^2) = \cos(\pi y x^2) \cdot \frac{\partial}{\partial x}(\pi y x^2) = \cos(\pi y x^2) \cdot 2\pi y x
\]
\[
f_x(0.5, 0.6) = \cos\left(\frac{3\pi}{16}\right) \cdot 2\pi \times 0.6 \times 0.5 = \cos\left(\frac{3\pi}{16}\right) \cdot 0.6\pi
\]
\[
\cos\left(\frac{3\pi}{16}\right) \approx 0.83147
\]
\[
f_x(0.5, 0.6) \approx 0.83147 \times 0.6\pi \approx 1.567
\]

**Partial derivative with respect to \( y \):**
\[
f_y(x, y) = \frac{\partial}{\partial y} \sin(\pi y x^2) = \cos(\pi y x^2) \cdot \frac{\partial}{\partial y}(\pi y x^2) = \cos(\pi y x^2) \cdot \pi x^2
\]
\[
f_y(0.5, 0.6) = \cos\left(\frac{3\pi}{16}\right) \cdot \frac{\pi}{4} \approx 0.83147 \times \frac{\pi}{4} \approx 0.6527
\]

### 3. **Apply the Taylor approximation:**
\[
f(0.7, 0.9) \approx f(0.5, 0.6) + f_x(0.5, 0.6) \cdot (0.7 - 0.5) + f_y(0.5, 0.6) \cdot (0.9 - 0.6)
\]
\[
f(0.7, 0.9) \approx 0.55557 + 1.567 \times 0.2 + 0.6527 \times 0.3
\]
\[
f(0.7, 0.9) \approx 0.55557 + 0.3134 + 0.19581 \approx 1.06
\]

### Final Answer:
\[
f(0.7, 0.9) \approx 1.06
\]

Would you like further details or have any questions? Here are some related questions you might consider:

1. How does the Taylor series change if the point of expansion is different?
2. What is the second-order Taylor approximation for this function?
3. How does the error in the approximation depend on the distance from the expansion point?
4. What would happen if we used a different function for the approximation?
5. Can we estimate \( f(0.7, 0.9) \) without using the partial derivatives?

**Tip:** When working with trigonometric functions in calculus, make sure to consistently use either radians or degrees throughout your calculations to avoid errors.

Learn how to estimate f(0.7, 0.9) using the first-order Taylor approximation for the function f(x, y) = sin(πyx^2). Includes step-by-step calculation of partial derivatives and application of the Taylor series. Suitable for advanced college-level mathematics.

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