Business Calculus Study Guidance: Derivatives, Limits, and Tangent Lines

Q1. Find and simplify: $\frac{d}{dx} \left( x^2 \cos x \right)$

Background

Topic: Derivatives (Product Rule)

This question tests your ability to differentiate a function that is the product of two functions: a polynomial and a trigonometric function.

Key Terms and Formulas

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Derivative of $x^2$: $\frac{d}{dx} x^2 = 2x$

• Derivative of $\cos x$: $\frac{d}{dx} \cos x = -\sin x$

Step-by-Step Guidance

1. Identify $f(x) = x^2$ and $g(x) = \cos x$.

2. Find $f'(x)$ and $g'(x)$: $f'(x) = 2x$, $g'(x) = -\sin x$.

3. Apply the product rule: $\frac{d}{dx} [x^2 \cos x] = f'(x)g(x) + f(x)g'(x)$.

4. Substitute the derivatives and functions into the formula.

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Q2. $\frac{d}{dx} \left( \sin^2 x \right)$

Background

Topic: Derivatives (Chain Rule)

This question tests your ability to differentiate a composite function, specifically the square of a trigonometric function.

Key Terms and Formulas

• Chain Rule: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

• Derivative of $\sin x$: $\frac{d}{dx} \sin x = \cos x$

Step-by-Step Guidance

1. Rewrite $\sin^2 x$ as $(\sin x)^2$ to clarify the composite structure.

2. Let $u = \sin x$, so the function is $u^2$.

3. Differentiate $u^2$ with respect to $u$: $\frac{d}{du} u^2 = 2u$.

4. Multiply by the derivative of $u$ with respect to $x$: $\frac{d}{dx} \sin x = \cos x$.

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Q3. $\frac{d}{dx} \left( \sqrt{x^2 + 1} \right)$

Background

Topic: Derivatives (Chain Rule)

This question tests your ability to differentiate a function involving a square root and a polynomial inside.

Key Terms and Formulas

• Chain Rule: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$

• Derivative of $\sqrt{u}$: $\frac{d}{du} \sqrt{u} = \frac{1}{2\sqrt{u}}$

• Derivative of $x^2 + 1$: $\frac{d}{dx} (x^2 + 1) = 2x$

Step-by-Step Guidance

1. Let $u = x^2 + 1$, so the function is $\sqrt{u}$.

2. Differentiate $\sqrt{u}$ with respect to $u$: $\frac{1}{2\sqrt{u}}$.

3. Multiply by the derivative of $u$ with respect to $x$: $\frac{d}{dx} (x^2 + 1) = 2x$.

4. Combine the results to express the derivative in terms of $x$.

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Q4. $\frac{d}{dx} \left( \frac{x^2 + 1}{x} \right)$

Background

Topic: Derivatives (Quotient Rule)

This question tests your ability to differentiate a rational function using the quotient rule.

Key Terms and Formulas

• Quotient Rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Derivative of $x^2 + 1$: $2x$

• Derivative of $x$: $1$

Step-by-Step Guidance

1. Let $f(x) = x^2 + 1$ and $g(x) = x$.

2. Find $f'(x) = 2x$ and $g'(x) = 1$.

3. Apply the quotient rule formula.

4. Substitute the derivatives and functions into the formula.

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Q5. $\lim_{x \to 0} \frac{\sin x}{x}$

Background

Topic: Limits (Trigonometric Limit)

This question tests your understanding of a fundamental trigonometric limit, which is important in calculus for evaluating derivatives of sine functions.

Key Terms and Formulas

• Limit: $\lim_{x \to 0} \frac{\sin x}{x}$

• This is a standard limit in calculus.

Step-by-Step Guidance

1. Recognize that direct substitution gives $\frac{0}{0}$, which is indeterminate.

2. Recall the standard trigonometric limit result for $\lim_{x \to 0} \frac{\sin x}{x}$.

3. Consider using L'Hôpital's Rule or the Taylor series expansion for $\sin x$ near $x = 0$.

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Q6. What change can you make to make $\frac{\sin x}{x}$ not undefined at $x = 0$?

Background

Topic: Limits and Continuity

This question tests your understanding of how to define a function at a point where it is otherwise undefined, using limits to ensure continuity.

Key Terms and Formulas

• Removable discontinuity: A point where a function is undefined but can be "fixed" by assigning a value based on the limit.

• Limit: $\lim_{x \to 0} \frac{\sin x}{x}$

Step-by-Step Guidance

1. Identify that $\frac{\sin x}{x}$ is undefined at $x = 0$ because the denominator is zero.

2. Consider what value you could assign to the function at $x = 0$ to make it continuous.

3. Use the limit as $x \to 0$ to determine the appropriate value.

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Q7. $\lim_{x \to 0} \frac{e^x - 1}{x}$

Background

Topic: Limits (Exponential Functions)

This question tests your ability to evaluate a limit involving an exponential function, which is important for understanding derivatives of exponential functions.

Key Terms and Formulas

• Limit: $\lim_{x \to 0} \frac{e^x - 1}{x}$

• L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Step-by-Step Guidance

1. Recognize that direct substitution gives $\frac{0}{0}$, which is indeterminate.

2. Apply L'Hôpital's Rule by differentiating the numerator and denominator.

3. Evaluate the new limit after differentiation.

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Q8. $f(x) = x^2$. Compute $f'(x)$ and $f'(2)$, and write the equation of the tangent line at $x = 2$.

Background

Topic: Derivatives and Tangent Lines

This question tests your ability to compute the derivative of a function, evaluate it at a specific point, and use it to write the equation of a tangent line.

Key Terms and Formulas

• Derivative: $f'(x)$

• Tangent line equation: $y = f'(a)(x - a) + f(a)$

Step-by-Step Guidance

1. Find $f'(x)$ for $f(x) = x^2$.

2. Evaluate $f'(2)$.

3. Find $f(2)$.

4. Set up the tangent line equation using $x = 2$.

Try solving on your own before revealing the answer!