Calculus Quiz 5: Limits, Derivatives, Linear Approximation, and Newton's Method

Q1. Use L’Hôpital’s Rule to compute the following limits:

1. $\lim\limits_{x \to 1} \frac{x^5 - x^3 - x^2 + 1}{x^6 - x^4 - x^2 + 1}$

2. $\lim\limits_{x \to \infty} \frac{x^2 \ln x - x}{x^2 + x \ln x}$

Background

Topic: Limits and L’Hôpital’s Rule

These questions test your ability to evaluate indeterminate limits using L’Hôpital’s Rule, which involves differentiating the numerator and denominator until the limit can be evaluated.

Key Terms and Formulas

• L’Hôpital’s Rule: If $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ yields an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:

  $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit on the right exists)

• Derivative of $x^n$: $\frac{d}{dx} x^n = n x^{n-1}$

• Derivative of $\ln x$: $\frac{d}{dx} \ln x = \frac{1}{x}$

Step-by-Step Guidance

1. For each limit, first check if direct substitution gives an indeterminate form ($\frac{0}{0}$ or $\frac{\infty}{\infty}$).

2. If so, differentiate the numerator and denominator separately with respect to $x$.

3. For the first limit, compute $f'(x)$ and $g'(x)$ for the numerator and denominator polynomials.

4. For the second limit, apply the product and chain rules as needed to differentiate terms involving $x^2 \ln x$ and $x \ln x$.

5. After differentiating, set up the new limit expressions. Do not evaluate the final limit yet.

Try solving on your own before revealing the answer!

Q2. Compute the derivative of $g(x) = \sqrt{x} - \sqrt[3]{x}$, find the linear approximation $L(x)$ to $g(x)$ at $a = 64$, and use $L(x)$ to approximate $g(65)$.

Background

Topic: Derivatives and Linear Approximation (Tangent Line Approximation)

This question tests your ability to compute derivatives, construct a linear (tangent line) approximation at a point, and use it to estimate function values near that point.

Key Terms and Formulas

• Derivative of $x^n$: $\frac{d}{dx} x^n = n x^{n-1}$

• Linear Approximation: $L(x) = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Rewrite $g(x)$ as $x^{1/2} - x^{1/3}$ to make differentiation easier.

2. Compute $g'(x)$ using the power rule for derivatives.

3. Evaluate $g(64)$ and $g'(64)$ to use in the linear approximation formula.

4. Write the linear approximation $L(x)$ using the formula $L(x) = g(64) + g'(64)(x - 64)$.

5. Set up the expression for $L(65)$ to approximate $g(65)$, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q3. Use 3 steps of Newton’s Method starting with $x_0 = 1$ to approximate a zero of $g(x) = x^4 + x - 1$. (Give the exact value of $x_1$ and use your calculator to compute $x_2$ and $x_3$ rounded to the fifth decimal place.)

Background

Topic: Newton’s Method for Approximating Roots

This question tests your understanding of Newton’s Method, an iterative technique for finding approximate solutions to equations of the form $g(x) = 0$.

Key Terms and Formulas

• Newton’s Method: $x_{n+1} = x_n - \frac{g(x_n)}{g'(x_n)}$

• Derivative of $g(x)$: For $g(x) = x^4 + x - 1$, $g'(x) = 4x^3 + 1$

Step-by-Step Guidance

1. Compute $g(x)$ and $g'(x)$ for the given function.

2. Set up the Newton’s Method formula for $x_1$ using $x_0 = 1$.

3. Calculate the exact value of $x_1$ (leave as a fraction or exact decimal).

4. Set up the expressions for $x_2$ and $x_3$ using the Newton’s Method formula, but do not compute their values yet.

Try solving on your own before revealing the answer!