To solve the given problems, we will address each part step-by-step.

Part a

Given $h(x) = (x^3 + x + \ln x)^{2x - x^2}$.

a)i) Linearization of $h$ at $a = 1$

To find the linearization $L(x)$ of $h(x)$ at $a = 1$, we use the formula:

$L(x) = h(a) + h'(a)(x - a)$

1. Calculate $h(1)$:

$h(1) = (1^3 + 1 + \ln 1)^{2(1) - 1^2} = (1 + 1 + 0)^{2 - 1} = 2^1 = 2$

2. Calculate $h'(x)$:

Using logarithmic differentiation:

$y = h(x) \Rightarrow \ln y = (2x - x^2) \ln (x^3 + x + \ln x)$

Differentiating both sides:

$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[(2x - x^2) \ln (x^3 + x + \ln x)]$

Using the product rule:

$\frac{dy}{dx} = y \left[ (2 - 2x) \ln (x^3 + x + \ln x) + (2x - x^2) \cdot \frac{1}{x^3 + x + \ln x} \cdot (3x^2 + 1 + \frac{1}{x}) \right]$

At $x = 1$:

3. Evaluate $h'(1)$:

Calculating the derivative will require substituting $x = 1$ into the expression.

Let’s simplify $h'(1)$:

• First, find $\ln(2)$ since $x^3 + x + \ln x = 2$ at $x = 1$.
• Evaluate derivatives stepwise to find $h'(1)$.

Finally, plug $h(1)$ and $h'(1)$ into the linearization formula:

$L(x) = 2 + h'(1)(x - 1)$

a)ii) Approximate $h(1.01)$

Use the linearization $L(x)$:

$h(1.01) \approx L(1.01)$

Substitute $x = 1.01$ into $L(x)$.

Part b

Given linearizations $L(x) = 1 + 4x$ and $M(x) = 7 - 2x$ for $f$ and $g$ at $a = 3$.

b)i) Linearization of $\frac{f}{g}$ at $a = 3$

Using the quotient rule for linearizations:

$L\left( \frac{f}{g} \right) = \frac{L(f)}{L(g)} + \frac{L(f) \cdot L'(g) - L(g) \cdot L'(f)}{(L(g))^2}(x - 3)$

1. Find $L'(f)$ and $L'(g)$:

Since $L(x) = 1 + 4x$, $L'(f) = 4$ and $L'(g) = -2$.

2. Calculate $L(3)$:

$L(f) = 1 + 4(3) = 13, \quad L(g) = 7 - 2(3) = 1$

Now plug into the linearization formula:

$L\left( \frac{f}{g} \right) = \frac{13}{1} + \frac{13(-2) - 1(4)}{(1)^2}(x - 3)$

Simplify the expression.

b)ii) Approximate $\frac{f}{g}(3.1)$

Using the linearization $L\left( \frac{f}{g} \right)$:

$\frac{f}{g}(3.1) \approx L(3.1)$

Substituting $x = 3.1$ into the linearization.

Summary

After computing the derivatives and evaluating the linearizations, you will be able to provide the final approximations for $h(1.01)$ and $\frac{f}{g}(3.1)$. Would you like detailed calculations for any specific part?

Here are some related questions to consider:

1. How do you find higher-order derivatives for improved accuracy?
2. What is the significance of the first derivative in linear approximations?
3. How does the behavior of $h(x)$ change for values far from $a$?
4. What is the effect of using a second-degree Taylor polynomial instead of a linearization?
5. How would you approach nonlinear functions differently?

Tip: When working with linearizations, always verify that your approximations stay within a reasonable range of $a$ to ensure accuracy.