Comprehensive Study Guidance for Calculus I Exam 2

Q1. Evaluate the limit $\lim_{x \to 0} \frac{\sin(2x)}{x}$

Background

Topic: Limits of Trigonometric Functions

This question tests your understanding of how to evaluate limits involving trigonometric functions, especially as $x$ approaches zero. It often requires knowledge of standard limit results and algebraic manipulation.

Key Terms and Formulas:

• Standard limit: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

• Substitution and properties of sine function

Step-by-Step Guidance

1. Recognize that the expression $\frac{\sin(2x)}{x}$ can be rewritten using substitution: let $u = 2x$.

2. Express the limit in terms of $u$: as $x \to 0$, $u \to 0$ as well.

3. Rewrite the limit: $\frac{\sin(2x)}{x} = 2 \cdot \frac{\sin(2x)}{2x}$.

4. Apply the standard limit result to $\frac{\sin(2x)}{2x}$ as $x \to 0$.

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Q2. Find the derivative of $f(x) = x^4 - 5x^2 + 7$

Background

Topic: Derivatives Using Formulae

This question tests your ability to differentiate polynomial functions using basic differentiation rules.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Apply the power rule to each term in the function $f(x)$.

2. Differentiate $x^4$ to get $4x^3$.

3. Differentiate $-5x^2$ to get $-10x$.

4. Differentiate the constant $7$to get $0$.

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Q3. Differentiate implicitly to find $\frac{dy}{dx}$ for $x^2 + y^2 = 25$

Background

Topic: Implicit Differentiation

This question tests your ability to use implicit differentiation to find derivatives when $y$ is defined implicitly as a function of $x$.

Key Terms and Formulas:

• Implicit differentiation: Differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$.

• Chain rule: $\frac{d}{dx} y^2 = 2y \frac{dy}{dx}$

Step-by-Step Guidance

1. Differentiate both sides of the equation $x^2 + y^2 = 25$ with respect to $x$.

2. Apply the derivative to $x^2$ to get $2x$.

3. Apply the derivative to $y^2$ using the chain rule to get $2y \frac{dy}{dx}$.

4. Set the derivative of the constant $25$to $0$.

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Q4. Find the equation of the tangent line to $f(x) = x^2 + 3x$ at $x = 2$

Background

Topic: Tangent Lines

This question tests your ability to find the equation of a tangent line to a curve at a specific point using derivatives.

Key Terms and Formulas:

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

• Derivative: $f'(x)$ gives the slope at any point $x$

Step-by-Step Guidance

1. Find $f'(x)$ for $f(x) = x^2 + 3x$ using the power rule.

2. Evaluate $f'(2)$ to get the slope at $x = 2$.

3. Find $f(2)$ to get the y-coordinate at $x = 2$.

4. Substitute $f'(2)$ and $f(2)$ into the tangent line formula.

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Q5. Differentiate $f(x) = \sin(2x + 1)$ using the chain rule

Background

Topic: Derivative by Chain Rule

This question tests your ability to apply the chain rule to differentiate composite functions.

Key Terms and Formulas:

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

• Derivative of $\sin(u)$: $\cos(u)$

Step-by-Step Guidance

1. Identify the inner function $u = 2x + 1$ and the outer function $f(u) = \sin(u)$.

2. Differentiate the outer function: $\frac{d}{du} \sin(u) = \cos(u)$.

3. Differentiate the inner function: $\frac{d}{dx} (2x + 1) = 2$.

4. Multiply the derivatives according to the chain rule.

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