Q1(a)(i). Find the limit:

Background

Topic: Limits and Trigonometric Functions

This question tests your understanding of evaluating limits involving trigonometric functions, especially as approaches 0. Recognizing standard limits and using algebraic manipulation is key.

Key Terms and Formulas

• Standard limit:

• Algebraic manipulation to match standard forms

Step-by-Step Guidance

1. Rewrite the limit to express both numerator and denominator in terms of .

2. Multiply and divide by appropriate constants to use the standard limit .

3. Express as .

4. Apply the standard limit to each part as .

Try solving on your own before revealing the answer!

Q1(a)(ii). Find the limit:

Background

Topic: Limits, Exponential Functions, and Indeterminate Forms

This question tests your ability to evaluate limits involving exponential functions and polynomials, especially when the direct substitution gives an indeterminate form (like ).

Key Terms and Formulas

• Exponential function:

• Standard limit:

• Factoring and simplifying expressions

Step-by-Step Guidance

1. Check what happens when you substitute (identify the indeterminate form).

2. Factor the denominator as .

3. Rewrite the numerator using the standard limit for .

4. Consider dividing numerator and denominator by to simplify.

Try solving on your own before revealing the answer!

Q1(b). State (but do not prove) the epsilon-delta definition of

Background

Topic: Formal Definition of a Limit (Epsilon-Delta)

This question tests your understanding of the precise (epsilon-delta) definition of a limit for a function as approaches a value.

Key Terms and Formulas

• Epsilon (): Arbitrarily small positive number representing how close is to the limit.

• Delta (): Corresponding positive number representing how close is to the point of interest.

Step-by-Step Guidance

1. Recall the general epsilon-delta definition for .

2. State the definition specifically for , , .

3. Express the definition using and .

Try stating the definition in your own words before checking the answer!

Q2. Use the Intermediate Value Theorem to show that has a solution in .

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the IVT, which guarantees the existence of a root for a continuous function on a closed interval if the function takes on values of opposite sign at the endpoints.

Key Terms and Formulas

• Intermediate Value Theorem: If is continuous on and and have opposite signs, then there exists such that .

• Define or for root finding.

Step-by-Step Guidance

1. Define a new function (or ).

2. Check that is continuous on (exponential and polynomial functions are continuous everywhere).

3. Compute and to determine their signs.

4. Verify that and have opposite signs.

Try applying the IVT before checking the answer!

Q3(a). Is there any relationship between the slope of the tangent line to a curve at a point and the derivative ? (Assume is defined at $a$.)

Background

Topic: Derivatives and Tangent Lines

This question tests your conceptual understanding of the geometric meaning of the derivative at a point.

Key Terms and Formulas

• Derivative: represents the instantaneous rate of change of at .

• Tangent line: The line that just touches the curve at a point and has the same slope as the curve at that point.

Step-by-Step Guidance

1. Recall the definition of the derivative at a point .

2. Relate the derivative to the slope of the tangent line at .

3. State the relationship in a clear sentence.

Try explaining the relationship in your own words before checking the answer!

Q3(b). For , find the equation of the tangent line at .

Background

Topic: Tangent Lines and Derivatives

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

Key Terms and Formulas

• Derivative:

• Point-slope form of a line:

Step-by-Step Guidance

1. Compute to find the -coordinate of the point of tangency.

2. Find , the derivative of .

3. Evaluate to get the slope of the tangent line at .

4. Write the equation of the tangent line using the point-slope form with and .

Try setting up the tangent line equation before checking the answer!

Q4. Suppose and are differentiable functions of , and . Use implicit differentiation to find . (No need to simplify.)

Background

Topic: Implicit Differentiation

This question tests your ability to use implicit differentiation to find when and are both functions of another variable (here, ).

Key Terms and Formulas

• Implicit differentiation: Differentiating both sides of an equation with respect to and solving for .

• Chain rule:

• Relationship:

Step-by-Step Guidance

1. Differentiate both sides of with respect to .

2. Apply the chain rule to each term: , , and .

3. Collect terms involving and .

4. Solve for using .

Try differentiating and setting up the expression before checking the answer!

Q5. Using the Bisection method, approximate with an error less than .

Background

Topic: Numerical Methods – Bisection Method

This question tests your understanding of the bisection method for approximating roots of equations to a specified accuracy.

Key Terms and Formulas

• Bisection method: A root-finding method that repeatedly bisects an interval and selects a subinterval in which a root must lie.

• Error bound: After steps, the error is at most for interval .

• Set up to find as the root.

Step-by-Step Guidance

1. Define and identify an interval where and have opposite signs.

2. Calculate the number of iterations needed so that the error is less than using .

Perform the first few bisection steps: compute midpoints and evaluate at those points.

1. Continue the process, updating the interval each time, until the error bound is met.

Try performing the bisection steps before checking the answer!