Q1. Find if .

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:

• Power Rule:

Step-by-Step Guidance

1. Rewrite as to make differentiation easier if you prefer the product/power rule.

2. Identify and for the quotient rule.

3. Compute and separately.

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

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Q2. For , find , at , and $y$ at .

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:

• Function Evaluation: Substitute the given value into .

Step-by-Step Guidance

1. Differentiate using the power rule to find .

2. To find at , substitute into and simplify.

3. To find at , substitute into and simplify.

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Q3. Refer to the graph of to describe the behavior of .

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: means the value approaches as approaches from the left.

Step-by-Step Guidance

1. Locate on the -axis of the graph.

2. Observe the -values as approaches $1 (from the left).

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

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Q4. Find .

Background

Topic: Limits of Linear Functions

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

Key Terms and Formulas

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

Step-by-Step Guidance

1. Recognize that is continuous everywhere, so you can use direct substitution.

2. Substitute into to set up the calculation.

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Q5. Use the graph of to estimate the following:

• a.

• b.

• c.

• d.

• e. Is continuous at ?

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: and

• Two-sided limit: exists if both one-sided limits exist and are equal.

• Continuity at : is continuous at if and both exist.

Step-by-Step Guidance

1. For each limit, use the graph to estimate the value approaches from the elsewhere specified direction.

2. For , check the value of the function at (look for the filled dot or defined value).

3. For continuity, check if the two-sided limit and exist and are equal.

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Q6. Sketch the graph of and determine where 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 for and for on the same axes.

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

3. Determine if the function is nondifferentiable at or elsewhere.

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