Business Calculus Exam 1 Review – Step-by-Step Guidance

Q1. Find the limit: $\displaystyle \lim_{x \to -3} \frac{x^2 - 5x - 24}{x^2 - 9}$

Background

Topic: Limits and Rational Functions

This question tests your ability to evaluate limits of rational functions, especially when direct substitution leads to an indeterminate form.

Key Terms and Formulas:

• Limit: The value a function approaches as the input approaches a certain value.

• Indeterminate Form: An expression like $\frac{0}{0}$, which requires algebraic manipulation to resolve.

Step-by-Step Guidance

1. Substitute $x = -3$ into the numerator and denominator to check if you get an indeterminate form.

2. If you get $0/0$, factor both the numerator and denominator to see if any common factors can be cancelled.

3. Simplify the expression by cancelling common factors.

4. After simplifying, substitute $x = -3$ into the new expression to evaluate the limit.

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Q2. Find the limit: $\displaystyle \lim_{x \to 2} \frac{3x^2 + x - 14}{x^2 - 7x + 10}$

Background

Topic: Limits and Rational Functions

This question is similar to Q1 and tests your ability to evaluate limits involving rational expressions, especially when substitution gives an indeterminate form.

Key Terms and Formulas:

• Limit and Indeterminate Form as above.

Step-by-Step Guidance

1. Substitute $x = 2$ into the numerator and denominator to check for an indeterminate form.

2. If you get $0/0$, factor both the numerator and denominator to look for common factors.

3. Simplify the expression by cancelling any common factors.

4. Substitute $x = 2$ into the simplified expression to find the limit.

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Q3. Find the derivative: $y = 4x^3 + 6x^2 + 9x - 7$

Background

Topic: Differentiation of Polynomials

This question tests your ability to apply the power rule to find the derivative of a polynomial function.

Key Terms and Formulas:

• Derivative: Measures the rate of change of a function.

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

Step-by-Step Guidance

1. Apply the power rule to each term of the polynomial separately.

2. For each term, multiply the coefficient by the exponent and decrease the exponent by one.

3. Combine the results to write the derivative function $y'$.

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Q4. Find the derivative: $f(x) = x^3 + 4x + 1$

Background

Topic: Differentiation of Polynomials

This question tests your ability to use the power rule for derivatives.

Key Terms and Formulas:

• Power Rule as above.

Step-by-Step Guidance

1. Differentiate each term using the power rule.

2. Remember that the derivative of a constant is zero.

3. Combine the results to write $f'(x)$.

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Q5. Find the derivative: $f(x) = 6x^3 + 3x^2 - 4x + 1$

Background

Topic: Differentiation of Polynomials

This question tests your ability to apply the power rule to a polynomial with both positive and negative coefficients.

Key Terms and Formulas:

• Power Rule as above.

Step-by-Step Guidance

1. Apply the power rule to each term in the function.

2. Be careful with negative signs and coefficients.

3. Combine the results to write $f'(x)$.

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Q6. Find the derivative: $g(x) = 4x^2 \cdot (2x + 5)^3$

Background

Topic: Product Rule and Chain Rule

This question tests your ability to use the product rule and chain rule to differentiate a product of two functions, one of which is a composite function.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Let $u(x) = 4x^2$ and $v(x) = (2x + 5)^3$.

2. Find $u'(x)$ and $v'(x)$ separately. For $v'(x)$, use the chain rule.

3. Apply the product rule to combine $u'(x)$, $v(x)$, $u(x)$, and $v'(x)$.

4. Simplify the resulting expression as much as possible.

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Q7. Find the derivative: $h(x) = x^5 + 4x^4 - x + 5$

Background

Topic: Differentiation of Polynomials

This question tests your ability to use the power rule for higher-degree polynomials.

Key Terms and Formulas:

• Power Rule as above.

Step-by-Step Guidance

1. Apply the power rule to each term in the polynomial.

2. Be careful with negative signs and constants.

3. Combine the results to write $h'(x)$.

Try solving on your own before revealing the answer!

Q8. Find the equation of the tangent line to $f(x) = x^2 - 2x$ at the point (4, 8).

Background

Topic: Tangent Lines and Derivatives

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

Key Terms and Formulas:

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

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Derivative: Gives the slope of the tangent line at a point.

Step-by-Step Guidance

1. Find $f'(x)$, the derivative of $f(x)$, to get the slope function.

2. Evaluate $f'(4)$ to find the slope of the tangent line at $x = 4$.

3. Use the point-slope form with the point (4, 8) and the slope from the previous step to write the equation of the tangent line.

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Q9. Consider the piecewise function: $f(x) = \begin{cases} x + 1 & \text{if } x < 4 \\ x + 5 & \text{if } x > 4 \end{cases}$

Background

Topic: Piecewise Functions and Limits

This question tests your understanding of limits and function values for piecewise-defined functions.

Key Terms and Formulas:

• Piecewise Function: A function defined by different expressions for different intervals of the domain.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

Step-by-Step Guidance

1. For (a): To find $\lim_{x \to 4^+} f(x)$, use the expression for $x > 4$ and substitute values approaching 4 from the right.

2. For (b): To find $\lim_{x \to 4^-} f(x)$, use the expression for $x < 4$ and substitute values approaching 4 from the left.

3. For (c): To find $f(4)$, check if the function is defined at $x = 4$ in either piece.

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Q10. Find the derivative of $f(x) = x^2 - 3x + 8$ using the limit definition of the derivative. Show each step carefully.

Background

Topic: Limit Definition of the Derivative

This question tests your understanding of the formal definition of the derivative and your ability to apply it step by step.

Key Terms and Formulas:

• Limit Definition of Derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Step-by-Step Guidance

1. Write out $f(x + h)$ by substituting $x + h$ into the function.

2. Compute $f(x + h) - f(x)$.

3. Set up the difference quotient $\frac{f(x + h) - f(x)}{h}$.

4. Simplify the numerator as much as possible before taking the limit as $h \to 0$.

Try solving on your own before revealing the answer!