(b) Determine an equation of the line tangent to the graph of at for the given value
of .
i)
ii)

Ask by Hobbs Christensen. in Kenya

Jan 20,2025

Question

(b) Determine an equation of the line tangent to the graph of at for the given value
of .
i)
ii)

Ask by Hobbs Christensen. in Kenya

Jan 20,2025

UpStudy AI · Accepted Answer

Sure, let's determine the equations of the tangent lines for both functions at the specified points.

---

### **i) $$ f(x) = \sqrt{3x + 1} $$ at $$ a = 8 $$**

**Step 1: Find $$ f(8) $$**
$$
f(8) = \sqrt{3(8) + 1} = \sqrt{24 + 1} = \sqrt{25} = 5
$$
So, the point of tangency is $$ (8, 5) $$.

**Step 2: Find the derivative $$ f'(x) $$**
$$
f(x) = (3x + 1)^{1/2} \
f'(x) = \frac{1}{2}(3x + 1)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x + 1}}
$$

**Step 3: Evaluate $$ f'(8) $$**
$$
f'(8) = \frac{3}{2\sqrt{3(8) + 1}} = \frac{3}{2\sqrt{25}} = \frac{3}{10}
$$
So, the slope $$ m = \frac{3}{10} $$.

**Step 4: Write the equation of the tangent line using the point-slope form**
$$
y - f(a) = m(x - a) \
y - 5 = \frac{3}{10}(x - 8)
$$
**Simplified Equation:**
$$
y = \frac{3}{10}x + \frac{13}{5}
$$

---

### **ii) $$ f(x) = \frac{2}{3x + 1} $$ at $$ a = -1 $$**

**Step 1: Find $$ f(-1) $$**
$$
f(-1) = \frac{2}{3(-1) + 1} = \frac{2}{-3 + 1} = \frac{2}{-2} = -1
$$
So, the point of tangency is $$ (-1, -1) $$.

**Step 2: Find the derivative $$ f'(x) $$**
$$
f(x) = 2(3x + 1)^{-1} \
f'(x) = 2 \cdot (-1)(3x + 1)^{-2} \cdot 3 = \frac{-6}{(3x + 1)^2}
$$

**Step 3: Evaluate $$ f'(-1) $$**
$$
f'(-1) = \frac{-6}{3(-1) + 1)^2} = \frac{-6}{(-3 + 1)^2} = \frac{-6}{4} = -\frac{3}{2}
$$
So, the slope $$ m = -\frac{3}{2} $$.

**Step 4: Write the equation of the tangent line using the point-slope form**
$$
y - f(a) = m(x - a) \
y - (-1) = -\frac{3}{2}(x - (-1)) \
y + 1 = -\frac{3}{2}(x + 1)
$$
**Simplified Equation:**
$$
y = -\frac{3}{2}x - \frac{5}{2}
$$

---

### **Final Answers:**

1. **For $$ f(x) = \sqrt{3x + 1} $$ at $$ a = 8 $$:**
   $$
   y = \frac{3}{10}x + \frac{13}{5}
   $$

2. **For $$ f(x) = \frac{2}{3x + 1} $$ at $$ a = -1 $$:**
   $$
   y = -\frac{3}{2}x - \frac{5}{2}
   $$
The tangent line equations are: 1. $$ y = \frac{3}{10}x + \frac{13}{5} $$ for $$ f(x) = \sqrt{3x + 1} $$ at $$ a = 8 $$. 2. $$ y = -\frac{3}{2}x - \frac{5}{2} $$ for $$ f(x) = \frac{2}{3x + 1} $$ at $$ a = -1 $$.

(b) Determine an equation of the line tangent to the graph of at for the given value
of .
i)
ii)

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Answer

Solution

i) at

ii) at

Final Answers:

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