Calculus I Test 1 Review – Step-by-Step Guidance

Q1. Find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P for $y = -5 - 5x^2$ at $P(3, -50)$.

Background

Topic: Derivatives and Tangent Lines

This question tests your ability to compute the derivative of a function (which gives the slope of the tangent line) and to use point-slope form to write the equation of the tangent line at a specific point.

Key Terms and Formulas

• Derivative: $y' = \frac{dy}{dx}$ gives the slope of the curve at any $x$.

• Point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope at $(x_1, y_1)$.

Step-by-Step Guidance

1. Find the derivative of $y = -5 - 5x^2$ with respect to $x$ to get the general formula for the slope.

2. Substitute $x = 3$ into your derivative to find the slope at $P(3, -50)$.

3. Use the point-slope form with $m$ (the slope you just found) and the point $(3, -50)$ to set up the equation of the tangent line.

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Q2. Which of the statements a through i about the function $f(x)$ graphed here are true, and which are false?

Background

Topic: Limits and Continuity from Graphs

This question tests your ability to interpret limits, function values, and continuity from a graph.

Key Terms and Concepts

• Limit: $\lim_{x \to c} f(x)$ is the value $f(x)$ approaches as $x$ approaches $c$.

• Function value: $f(c)$ is the actual value of the function at $x = c$.

• Continuity: $f$ is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$.

Step-by-Step Guidance

1. For each statement, carefully examine the graph at the specified $x$-value to determine if the limit exists and/or what the function value is.

2. Recall that a limit exists at $x = c$ if the left and right limits are equal.

3. Check if the function is defined at the point and if the limit equals the function value for continuity.

4. Mark each statement as true or false based on your analysis, but do not fill in the final answers yet.

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Q3. Find the following limit: $\lim_{x \to 5} \frac{-x^2 + 8x - 1}{2}$

Background

Topic: Evaluating Limits of Rational Functions

This question tests your ability to evaluate limits by direct substitution when the function is continuous at the point.

Key Terms and Formulas

• Limit: $\lim_{x \to a} f(x)$

• Direct substitution: If $f(x)$ is continuous at $x = a$, substitute $x = a$ directly.

Step-by-Step Guidance

1. Check if the denominator is zero at $x = 5$. Here, the denominator is a constant, so it's not zero.

2. Substitute $x = 5$ into the numerator $-x^2 + 8x - 1$.

3. Divide the result by $2$ to get the limit value.

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Q4. Find the limit: $\lim_{x \to -1} \frac{x + 1}{x^2 + 7x + 6}$

Background

Topic: Limits Involving Factorization

This question tests your ability to evaluate limits where substitution gives $0/0$ and you may need to factor and simplify.

Key Terms and Formulas

• Indeterminate form: $0/0$ suggests factoring or simplifying.

• Factoring quadratics: $x^2 + 7x + 6 = (x + 1)(x + 6)$

Step-by-Step Guidance

1. Substitute $x = -1$ into the numerator and denominator to check if you get $0/0$.

2. If you get $0/0$, factor the denominator and see if you can cancel a common factor with the numerator.

3. After simplifying, substitute $x = -1$ into the simplified expression to find the limit.

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Q5. Find the limit: $\lim_{x \to 5} \frac{x^2 - 2x - 15}{x - 5}$

Background

Topic: Limits and Removable Discontinuities

This question tests your ability to handle limits where direct substitution gives $0/0$ and you need to factor and simplify.

Key Terms and Formulas

• Removable discontinuity: Occurs when a factor cancels in numerator and denominator.

• Factoring: $x^2 - 2x - 15 = (x - 5)(x + 3)$

Step-by-Step Guidance

1. Substitute $x = 5$ into the numerator and denominator to check for $0/0$.

2. Factor the numerator and cancel the common factor with the denominator.

3. Substitute $x = 5$ into the simplified expression to find the limit.

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Q6. Find the limit: $\lim_{x \to 1} \frac{x - x^2}{1 - x}$

Background

Topic: Limits and Simplification

This question tests your ability to manipulate algebraic expressions to evaluate limits, especially when you get an indeterminate form.

Key Terms and Formulas

• Indeterminate form: $0/0$ suggests simplification is needed.

• Factoring: $x - x^2 = x(1 - x)$

Step-by-Step Guidance

1. Substitute $x = 1$ to check if you get $0/0$.

2. Factor the numerator and see if you can cancel with the denominator.

3. Substitute $x = 1$ into the simplified expression to find the limit.

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Q7. Find the limit: $\lim_{x \to 3} \frac{x^2 + 40 - 7x}{x - 3}$

Background

Topic: Limits and Factorization

This question tests your ability to factor and simplify rational expressions to evaluate limits at points of removable discontinuity.

Key Terms and Formulas

• Removable discontinuity: Factor numerator to see if $(x - 3)$ cancels.

Step-by-Step Guidance

1. Substitute $x = 3$ to check for $0/0$.

2. Factor the numerator and cancel with the denominator if possible.

3. Substitute $x = 3$ into the simplified expression to find the limit.

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Q8. Find the right-hand and left-hand limits for $f(x) = |x + 9| + 11$ at $c = -9$.

Background

Topic: One-Sided Limits and Absolute Value Functions

This question tests your understanding of one-sided limits and how absolute value affects the function's behavior at a point.

Key Terms and Formulas

• Right-hand limit: $\lim_{x \to c^+} f(x)$

• Left-hand limit: $\lim_{x \to c^-} f(x)$

• Absolute value: $|x|$ is $x$ if $x \geq 0$, $-x$ if $x < 0$

Step-by-Step Guidance

1. For $x \to -9^+$, consider values just greater than $-9$ and evaluate $|x + 9|$.

2. For $x \to -9^-$, consider values just less than $-9$ and evaluate $|x + 9|$.

3. Add $11$ to each result to find the one-sided limits.

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Q9. Find the limit: $\lim_{x \to 0} x \csc(8x) \cos(2x)$ using $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$.

Background

Topic: Trigonometric Limits

This question tests your ability to manipulate trigonometric expressions and use standard limits involving sine and cosine.

Key Terms and Formulas

• $\csc(8x) = \frac{1}{\sin(8x)}$

• Standard limit: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

Step-by-Step Guidance

1. Rewrite $x \csc(8x) \cos(2x)$ as $x \cdot \frac{1}{\sin(8x)} \cdot \cos(2x)$.

2. Combine $x$ and $\sin(8x)$ to form a ratio similar to the standard limit.

3. Use substitution or algebraic manipulation to express the limit in terms of $\frac{\sin(8x)}{8x}$ or its reciprocal.

4. Apply the standard limit and evaluate the remaining factors, but stop before the final calculation.

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Q10. Use the relation $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ to determine $\lim_{x \to 0} \frac{2x + 2x \cos(2x)}{5 \sin(2x) \cos(2x)}$.

Background

Topic: Trigonometric Limits and Algebraic Manipulation

This question tests your ability to use trigonometric identities and standard limits to evaluate more complex expressions.

Key Terms and Formulas

• Trigonometric identities: $1 + \cos(2x) = 2\cos^2(x)$, $\sin(2x) = 2\sin(x)\cos(x)$

• Standard limit: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

Step-by-Step Guidance

1. Factor $2x$ from the numerator: $2x(1 + \cos(2x))$.

2. Express the denominator in terms of $\sin(2x)$ and $\cos(2x)$.

3. Rewrite the expression to match the standard limit form if possible.

4. Apply the standard limit and simplify, but do not compute the final value yet.

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