Give step-by-step solution with explanation and final answer:Question 6 Let f(x,y) = e* Vcos(z — 1). Estimate £(0.98,3.01) using differentials (linear approximation)
Question 7 Change the following t

Question

Give step-by-step solution with explanation and final answer:Question 6 Let f(x,y) = e* Vcos(z — 1). Estimate £(0.98,3.01) using differentials (linear approximation)
Question 7 Change the following triple integral to cylindrical coordinates and then to spherical coordinates:
3 VIR EER
/ / / 2a? +y? +22 dzdydz. Use one of the three integrals to compute the common value
slve=mlo
Question 8 Evaluate f arctan(z) dz + (3z — 4 — 5y) dy, where C is the circle of radius 4 centered
c
at (2,5) parameterized counterclockwise. (Note. The formula for area of a circle may help, if you show how.)
Question 9 The fluid flow in a region in the plane is given by a vector field F = (z — 2y)i + (22 + 6y)j
Compute the total outward flux of the fluid passing through a rectangular box, with opposite corners at the
origin and at (4,2). Is there more flow into or out of the box? Note. Our present text seems a bit tight-lipped
regarding well-accepted terminology. The integral fF - Tds is usually called work integral or flow integral, try to
memorize "flow along” a curve C, as we take the component of the vector field F that goes with the tangent vector T.
If Cis closed then the same integral is

VIPSolutions · Accepted Answer

Let's solve **Question 6** step by step:

---

### **Question 6**
Let $ f(x,y) = e^{3x-y} \cos(x-1) $. Estimate $ f(0.98, 3.01) $ using differentials (linear approximation).

---

### **Step 1: Find the linear approximation formula**

The linear approximation (differential) for a function $ f(x,y) $ near the point $ (a,b) $ is:

$$
f(x, y) \approx f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
$$

where $ f_x $ and $ f_y $ are the partial derivatives with respect to $ x $ and $ y $.

---

### **Step 2: Define $ a $ and $ b $**

Let $ (a, b) = (1, 3) $, since $ (0.98, 3.01) $ is close to $ (1, 3) $.

Let $ (x, y) = (0.98, 3.01) $.

So,

$$
\Delta x = x - a = 0.98 - 1 = -0.02
$$
$$
\Delta y = y - b = 3.01 - 3 = 0.01
$$

---

### **Step 3: Compute $ f(a, b) $**

$$
f(1, 3) = e^{3 \cdot 1 - 3} \cos(1-1) = e^{0} \cos(0) = 1 \cdot 1 = 1
$$

---

### **Step 4: Compute the partial derivatives**

#### **Partial with respect to $ x $:**

$$
f_x(x, y) = \frac{\partial}{\partial x} \left( e^{3x-y} \cos(x-1) \right)
$$

Using the product rule:

$$
= \frac{\partial}{\partial x} e^{3x-y} \cdot \cos(x-1) + e^{3x-y} \cdot \frac{\partial}{\partial x} \cos(x-1)
$$
$$
= 3e^{3x-y} \cos(x-1) - e^{3x-y} \sin(x-1)
$$
$$
= e^{3x-y} \left[ 3\cos(x-1) - \sin(x-1) \right]
$$

Evaluate at $ (1, 3) $:

$$
f_x(1, 3) = e^{0} [3\cos(0) - \sin(0)]
= 1 \cdot [3 \cdot 1 - 0] = 3
$$

---

#### **Partial with respect to $ y $:**

$$
f_y(x, y) = \frac{\partial}{\partial y} (e^{3x-y} \cos(x-1))
$$
$$
= -e^{3x-y} \cos(x-1)
$$

Evaluate at $ (1, 3) $:

$$
f_y(1, 3) = -e^{0} \cos(0) = -1 \cdot 1 = -1
$$

---

### **Step 5: Apply the linear approximation**

$$
f(0.98, 3.01) \approx f(1, 3) + f_x(1, 3) \Delta x + f_y(1, 3) \Delta y
$$
$$
= 1 + 3 \times (-0.02) + (-1) \times 0.01
$$
$$
= 1 - 0.06 - 0.01
$$
$$
= 0.93
$$

---

## **Final Answer**

$$
\boxed{f(0.98, 3.01) \approx 0.93}
$$

This is the estimate using the differential (linear approximation) method.

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