Business Calculus Chapter 2 Review – Step-by-Step Study Guidance

Q1. Find $\frac{dy}{dx}$ if $y = \frac{4x-9}{36x^3}$.

Background

Topic: Differentiation of Rational Functions

This question tests your ability to differentiate a function that is a quotient of two polynomials. You may use the quotient rule or rewrite the function to use the power rule.

Key Terms and Formulas

• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

• Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$

Step-by-Step Guidance

1. Rewrite $y$ as $y = (4x-9) \cdot (36x^3)^{-1}$ to make differentiation easier if you prefer the product/power rule.

2. Identify $u = 4x-9$ and $v = 36x^3$ for the quotient rule.

3. Compute $\frac{du}{dx}$ and $\frac{dv}{dx}$ separately.

4. Apply the quotient rule formula to combine your results.

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Q2. For $y = 5x^4$, find $\frac{dy}{dx}$, $y$ at $x_1 = 3$, and $y$ at $x_2 = 6$.

Background

Topic: Power Rule and Function Evaluation

This question asks you to differentiate a power function and evaluate the function at specific points.

Key Terms and Formulas

• Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$

• Function Evaluation: Substitute the given $x$ value into $y$.

Step-by-Step Guidance

1. Differentiate $y = 5x^4$ using the power rule to find $\frac{dy}{dx}$.

2. To find $y$ at $x_1 = 3$, substitute $x = 3$ into $y = 5x^4$ and simplify.

3. To find $y$ at $x_2 = 6$, substitute $x = 6$ into $y = 5x^4$ and simplify.

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Q3. Refer to the graph of $y = f(x)$ to describe the behavior of $\lim_{x \to 1^-} f(x)$.

Background

Topic: One-Sided Limits from a Graph

This question tests your understanding of how to estimate one-sided limits using a graph.

Key Terms and Formulas

• One-sided limit: $\lim_{x \to a^-} f(x)$ means the value $f(x)$ approaches as $x$ approaches $a$ from the left.

Step-by-Step Guidance

1. Locate $x = 1$ on the $x$-axis of the graph.

2. Observe the $y$-values as $x$ approaches $1$from values less than $1$ (from the left).

3. Estimate the value that $f(x)$ is approaching as $x$ gets closer to $1$ from the left.

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Q4. Find $\lim_{x \to -6} (11x + 2)$.

Background

Topic: Limits of Linear Functions

This question tests your ability to evaluate the limit of a linear function as $x$ approaches a specific value.

Key Terms and Formulas

• Direct Substitution: For polynomials and linear functions, the limit as $x$ approaches $a$ is simply the function evaluated at $a$.

Step-by-Step Guidance

1. Recognize that $11x + 2$ is continuous everywhere, so you can use direct substitution.

2. Substitute $x = -6$ into $11x + 2$ to set up the calculation.

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Q5. Use the graph of $f$ to estimate the following:

• a. $\lim_{x \to 1^-} f(x)$

• b. $\lim_{x \to 1^+} f(x)$

• c. $\lim_{x \to 1} f(x)$

• d. $f(1)$

• e. Is $f$ continuous at $x = 1$?

Background

Topic: Limits and Continuity from a Graph

This question asks you to estimate one-sided and two-sided limits, function values, and continuity using a graph.

Key Terms and Formulas

• One-sided limits: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$

• Two-sided limit: $\lim_{x \to a} f(x)$ exists if both one-sided limits exist and are equal.

• Continuity at $x = a$: $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$ and both exist.

Step-by-Step Guidance

1. For each limit, use the graph to estimate the value $f(x)$ approaches from the elsewhere specified direction.

2. For $f(1)$, check the value of the function at $x = 1$ (look for the filled dot or defined value).

3. For continuity, check if the two-sided limit and $f(1)$ exist and are equal.

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Q6. Sketch the graph of $f(x) = \begin{cases} -4x & x < 1 \\ -4 & x \geq 1 \end{cases}$ and determine where $f$ is nondifferentiable.

Background

Topic: Piecewise Functions and Differentiability

This question tests your understanding of graphing piecewise functions and identifying points of nondifferentiability (such as corners or jumps).

Key Terms and Formulas

• Piecewise function: Defined by different expressions on different intervals.

• Nondifferentiable point: Where the function has a sharp corner, cusp, or discontinuity.

Step-by-Step Guidance

1. Sketch $y = -4x$ for $x < 1$ and $y = -4$ for $x \geq 1$ on the same axes.

2. Check the point $x = 1$ for a corner or discontinuity by evaluating the left and right derivatives.

3. Determine if the function is nondifferentiable at $x = 1$ or elsewhere.

Try solving on your own before revealing the answer!