Step-by-Step Calculus Guidance: Limits and Derivatives

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Q1a. Evaluate the limit:

Background

Topic: Limits of Trigonometric Functions

This question tests your understanding of how to evaluate the limit of a trigonometric function as the variable approaches zero.

Key Terms and Formulas

Limit: The value a function approaches as the input approaches a certain value.
Key limit property: for any constant .

Step-by-Step Guidance

Recognize that as , as well.
Recall the basic limit: .
Apply this property to , noting the argument is instead of .

Try solving on your own before revealing the answer!

Final Answer: $0$

As \sin(4x)$ approaches $0$ because the sine of $0.

Q1b. Evaluate the limit:

Background

Topic: Limits and Trigonometric Identities

This question tests your ability to use trigonometric identities and evaluate limits involving squared sine and cosine functions.

Key Terms and Formulas

Trigonometric identity: for any .
Limit properties for continuous functions.

Step-by-Step Guidance

Evaluate as . Recall .
Evaluate as . Recall .
Add the two results together to find the limit.

Try solving on your own before revealing the answer!

Final Answer: $1$

and as , so the sum is $1$.

Q2. Evaluate the limit:

Background

Topic: Limits Involving Products of Trigonometric Functions

This question tests your ability to evaluate limits involving products of trigonometric functions as approaches zero.

Key Terms and Formulas

Key limits: and

Step-by-Step Guidance

Rewrite as .
Express the product: .
As , and , but consider the rate at which they approach zero.
Set up the limit using the small angle approximation or L'Hospital's Rule if needed.

Try solving on your own before revealing the answer!

Final Answer: $0$

Both and approach $0x \to 0.

Q3. Find the derivative of the function:

Background

Topic: Product Rule for Differentiation

This question tests your ability to differentiate a product of two functions using the product rule.

Key Terms and Formulas

Product Rule:
Derivative of :

Step-by-Step Guidance

Let and .
Compute and separately.
Apply the product rule: .
Substitute the derivatives and simplify the expression.

Try solving on your own before revealing the answer!

Final Answer:

We used the product rule and the chain rule for the derivative of .

Q4. Find the derivative of the function:

Background

Topic: Chain Rule and Power Rule for Differentiation

This question tests your ability to differentiate composite trigonometric functions raised to a power.

Key Terms and Formulas

Chain Rule:
Power Rule:
Derivatives: ,

Step-by-Step Guidance

For , use the chain rule and power rule: .
For , use the chain rule and power rule: .
Combine the derivatives for the final expression.

Try solving on your own before revealing the answer!

Final Answer:

Each term was differentiated using the chain rule and power rule.

Q5. Find the derivative of the function:

Background

Topic: Product Rule and Chain Rule for Differentiation

This question tests your ability to differentiate a product involving a composite trigonometric function.

Key Terms and Formulas

Product Rule:
Chain Rule for nested functions
Derivatives:

Step-by-Step Guidance

Let and .
Find and using the chain rule for .
Apply the product rule: .
Set up the derivative for by differentiating step by step.

Try solving on your own before revealing the answer!

Final Answer:

The chain rule was applied multiple times for the composite function.

Q6. Find the derivative of the function:

Background

Topic: Derivatives of Trigonometric Functions

This question tests your knowledge of the derivatives of secant and tangent functions, and the product rule.

Key Terms and Formulas

Derivative of :
Derivative of :
Product Rule:

Step-by-Step Guidance

Let and .
Find and using the known derivatives.
Apply the product rule to combine the derivatives.
Simplify the resulting expression.

Try solving on your own before revealing the answer!

Final Answer:

We used the product rule and the derivatives of secant and tangent.

Q7. Find the derivative of the function:

Background

Topic: Product Rule and Chain Rule for Differentiation

This question tests your ability to differentiate a product involving a trigonometric function raised to a power.

Key Terms and Formulas

Derivative of :
Derivative of :
Product Rule:

Step-by-Step Guidance

Let and .
Find and using the chain rule.
Apply the product rule to .
Remember to differentiate the constant term $1$ as well.

Try solving on your own before revealing the answer!

Final Answer:

We applied the product rule and chain rule to differentiate the expression.

Q8. Evaluate the limit:

Background

Topic: Limits Involving Trigonometric Functions and Small Angle Approximations

This question tests your ability to use standard trigonometric limits and properties as approaches zero.

Key Terms and Formulas

Key limit:

Step-by-Step Guidance

Recognize the standard limit form as .
Recall that approaches $1x \to 0$.
Multiply the two limits together to set up the final calculation.

Try solving on your own before revealing the answer!

Final Answer: $3$

and as , so the product is $3$.

Q9. Find the derivative of the function:

Background

Topic: Product Rule and Derivative of Cotangent

This question tests your ability to differentiate a product involving a power function and the cotangent function.

Key Terms and Formulas

Product Rule:
Derivative of :
Derivative of :

Step-by-Step Guidance

Let and .
Find and using the known derivatives.
Apply the product rule to combine the derivatives.
Simplify the resulting expression.

Try solving on your own before revealing the answer!

Final Answer:

We used the product rule and the derivative of cotangent.