Math 140 (Calculus I) Final Exam Study Guidance

Q1. Use the graph to find the following limits and function values:

• a) $\lim_{x \to 2^-} f(x)$

• b) $\lim_{x \to 2^+} f(x)$

• c) $\lim_{x \to 2} f(x)$

• d) $f(2)$

Background

Topic: Limits and Function Values from Graphs

This question tests your ability to interpret a graph to determine left-hand and right-hand limits, the two-sided limit, and the actual function value at a point.

Key Terms:

• Left-hand limit: $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right.

• Two-sided limit: $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

• Function value: $f(a)$ is the actual value of the function at $x = a$ (may differ from the limit).

Step-by-Step Guidance

1. Examine the graph near $x = 2$. Identify the $y$-value that $f(x)$ approaches as $x$ approaches $2$from the left ($x \to 2^-$).

2. Identify the $y$-value that $f(x)$ approaches as $x$ approaches $2$from the right ($x \to 2^+$).

3. Compare the left-hand and right-hand limits to determine if the two-sided limit $\lim_{x \to 2} f(x)$ exists.

4. Find the actual value of $f(2)$ by looking for a closed or open dot at $x = 2$ on the graph.

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Q2. Evaluate the limit: $\lim_{h \to 0} \frac{\sqrt{97 + h} - \sqrt{97}}{h}$

Background

Topic: Limits Involving Radicals (Difference Quotient)

This question tests your ability to evaluate a limit that resembles the definition of the derivative, often requiring rationalization.

Key Terms and Formulas:

• Difference quotient: $\frac{f(a+h) - f(a)}{h}$

• Rationalization: Multiply numerator and denominator by the conjugate to simplify expressions with radicals.

Step-by-Step Guidance

1. Recognize that the numerator is a difference of square roots. The limit is indeterminate in the form $0/0$ as $h \to 0$.

2. Multiply numerator and denominator by the conjugate: $\sqrt{97 + h} + \sqrt{97}$.

3. Simplify the numerator using the difference of squares: $[\sqrt{97 + h} - \sqrt{97}][\sqrt{97 + h} + \sqrt{97}] = (97 + h) - 97 = h$.

4. Rewrite the expression and simplify as much as possible before substituting $h = 0$.

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Q3. Let $f(x) = \begin{cases} x^2 + 1, & x < -10 \\ x + 10, & x \geq -10 \end{cases}$. Compute the following limits or show that they do not exist:

• a) $\lim_{x \to -10^-} f(x)$

• b) $\lim_{x \to -10^+} f(x)$

• c) $\lim_{x \to -10} f(x)$

Background

Topic: Limits of Piecewise Functions

This question tests your understanding of how to evaluate limits for piecewise-defined functions, especially at the point where the formula changes.

Key Terms:

• Piecewise function: A function defined by different expressions on different intervals.

• One-sided limits: Use the appropriate piece of the function for $x$ approaching from the left or right.

Step-by-Step Guidance

1. For $x \to -10^-$, use the formula $f(x) = x^2 + 1$ and substitute $x = -10$.

2. For $x \to -10^+$, use the formula $f(x) = x + 10$ and substitute $x = -10$.

3. Compare the left and right limits to determine if the two-sided limit exists at $x = -10$.

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Q4. Find the derivative:

• a) $h(x) = 4x^2 + 3x^2 - 5$

• b) $y = 4x^5 \sin x$

• c) $y = \tan(9x^4)$

Background

Topic: Differentiation Rules (Sum, Product, Chain Rule)

This question tests your ability to apply basic differentiation rules, including the product and chain rules.

Key Formulas:

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

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

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

• Derivative of $\sin x$: $\cos x$

• Derivative of $\tan x$: $\sec^2 x$

Step-by-Step Guidance

1. For each part, identify which differentiation rule(s) apply (power, product, chain).

2. Apply the rule(s) step by step, showing the derivative of each term.

3. For part (b), use the product rule for $4x^5 \sin x$.

4. For part (c), use the chain rule for $\tan(9x^4)$.

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Q5. Let $f(x) = 3x^2 + 5x$. Find an equation for the line tangent to the graph of $f(x)$ at $(-2, 2)$.

Background

Topic: Tangent Lines to Curves

This question tests your ability to find the equation of a tangent line using derivatives and point-slope form.

Key Formulas:

• Derivative: $f'(x)$ gives the slope of the tangent line at $x$.

• Point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Find $f'(x)$ by differentiating $f(x)$.

2. Evaluate $f'(-2)$ to get the slope at $x = -2$.

3. Use the point $(-2, 2)$ and the slope in the point-slope form to write the equation of the tangent line.

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Q6. Suppose $s(t) = t^3 - 9t^2 + 24t$ measures the position of an object moving horizontally for $0 \leq t \leq 6$ where $s$ is in feet and $t$ in seconds.

• a) Find and graph the velocity function $v(t)$.

• b) Determine when the object is stationary, moving left, and moving right. Use interval notation.

Background

Topic: Velocity and Motion Analysis

This question tests your ability to find velocity as the derivative of position and analyze intervals of motion.

Key Formulas:

• Velocity: $v(t) = s'(t)$

• Stationary: $v(t) = 0$

• Moving left/right: $v(t) < 0$ (left), $v(t) > 0$ (right)

Step-by-Step Guidance

1. Differentiate $s(t)$ to find $v(t)$.

2. Solve $v(t) = 0$ to find when the object is stationary.

3. Test intervals between these points to determine where $v(t)$ is positive (moving right) or negative (moving left).

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Q7. Use implicit differentiation to find $\frac{dy}{dx}$ for $5x^4 + 3y^2 = 9xy$.

Background

Topic: Implicit Differentiation

This question tests your ability to differentiate equations involving both $x$ and $y$ and solve for $\frac{dy}{dx}$.

Key Steps:

• Differentiating both sides with respect to $x$.

• Remembering to use the chain rule for terms involving $y$ (treat $y$ as a function of $x$).

Step-by-Step Guidance

1. Differentiate each term with respect to $x$, applying the chain rule to $y^2$ and the product rule to $9xy$.

2. Collect all terms involving $\frac{dy}{dx}$ on one side of the equation.

3. Factor and solve for $\frac{dy}{dx}$.

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Q8. A spherical balloon is inflating with helium at a rate of $800\pi$ ft$^3$/min. How fast is the balloon’s radius increasing when the radius is 5 ft? (Hint: $V = \frac{4}{3}\pi r^3$)

Background

Topic: Related Rates

This question tests your ability to relate rates of change using implicit differentiation and the geometry of a sphere.

Key Formulas:

• Volume of a sphere: $V = \frac{4}{3}\pi r^3$

• Related rates: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$

Step-by-Step Guidance

1. Differentiate both sides of $V = \frac{4}{3}\pi r^3$ with respect to $t$ to relate $\frac{dV}{dt}$ and $\frac{dr}{dt}$.

2. Plug in the given values: $\frac{dV}{dt} = 800\pi$ and $r = 5$.

3. Solve for $\frac{dr}{dt}$, the rate at which the radius is increasing.

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Q9. Let $f(x) = 2x^3 + 3x^2 - 120x + 3$.

• a) Find $f'(x)$ and the critical points of $f(x)$.

• b) Use the first derivative test to identify the $x$-values of any local maxima and minima of $f(x)$ (if any).

• c) On the closed interval $[-6, 10]$ identify the absolute maxima and minima of $f(x)$.

Background

Topic: Critical Points, First Derivative Test, Absolute Extrema

This question tests your ability to find and classify critical points and determine absolute extrema on a closed interval.

Key Steps:

• Find $f'(x)$ and set it to zero to find critical points.

• Use the first derivative test to classify each critical point.

• Evaluate $f(x)$ at critical points and endpoints to find absolute extrema.

Step-by-Step Guidance

1. Differentiate $f(x)$ to find $f'(x)$.

2. Solve $f'(x) = 0$ to find critical points.

3. Use sign analysis or the first derivative test to classify each critical point as a local maximum or minimum.

4. Evaluate $f(x)$ at the critical points and at $x = -6$ and $x = 10$ to compare values for absolute extrema.

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Q10. Let $f(x) = 3x^4 - 2x^3 + 8$.

• a) Find $f''(x)$ and determine where $f(x)$ is concave up or concave down. Use interval notation.

• b) Identify the $x$-values of any inflection points.

Background

Topic: Concavity and Inflection Points

This question tests your ability to use the second derivative to determine concavity and locate inflection points.

Key Steps:

• Find $f''(x)$ by differentiating twice.

• Solve $f''(x) = 0$ to find possible inflection points.

• Test intervals to determine where $f(x)$ is concave up ($f''(x) > 0$) or concave down ($f''(x) < 0$).

Step-by-Step Guidance

1. Find $f'(x)$ and then $f''(x)$.

2. Solve $f''(x) = 0$ for $x$ to find candidates for inflection points.

3. Test values in the intervals determined by these $x$-values to see where $f''(x)$ is positive or negative.

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Q11. Evaluate the limit. Use L’Hôpital’s Rule when necessary.

• a) $\lim_{x \to 6^-} \frac{x^3 + 5x^2 + 6x}{x^2 - 36}$

• b) $\lim_{x \to 0} \frac{1 - \cos 7x^3}{x^2}$

Background

Topic: Limits and L’Hôpital’s Rule

This question tests your ability to evaluate indeterminate forms using L’Hôpital’s Rule.

Key Steps:

• Check if the limit is in an indeterminate form ($0/0$ or $\infty/\infty$).

• If so, differentiate numerator and denominator separately and try the limit again.

Step-by-Step Guidance

1. Substitute the value to check for indeterminate form.

2. If indeterminate, apply L’Hôpital’s Rule: differentiate numerator and denominator.

3. Repeat as necessary until the limit can be evaluated.

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Q12. Find $f(x)$ if $f'(x) = 12x - 5$ and $f(0) = 5$.

Background

Topic: Antiderivatives and Initial Conditions

This question tests your ability to find an original function given its derivative and an initial value.

Key Steps:

• Integrate $f'(x)$ to find $f(x)$.

• Use the initial condition $f(0) = 5$ to solve for the constant of integration.

Step-by-Step Guidance

1. Integrate $12x - 5$ with respect to $x$ to get $f(x)$ plus a constant $C$.

2. Plug in $x = 0$ and $f(0) = 5$ to solve for $C$.

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Q13. Evaluate the integral $\int_0^1 x^2 - 3x + 7\ dx$

Background

Topic: Definite Integrals

This question tests your ability to compute a definite integral using the Fundamental Theorem of Calculus.

Key Steps:

• Find the antiderivative of the integrand.

• Evaluate the antiderivative at the upper and lower limits and subtract.

Step-by-Step Guidance

1. Find the antiderivative $F(x)$ of $x^2 - 3x + 7$.

2. Compute $F(1) - F(0)$.

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Q14. Evaluate the following using the Fundamental Theorem of Calculus: $\frac{d}{dx} \int_0^{x^2} (t^2 + 13) dt$

Background

Topic: Fundamental Theorem of Calculus (Part 1 and Chain Rule)

This question tests your ability to differentiate an integral with a variable upper limit.

Key Steps:

• Recall: $\frac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Identify $f(t)$ and $g(x)$ in the integral.

2. Apply the chain rule as stated above.

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Q15. Evaluate the integral.

• a) $\int \sin(7x + 2) dx$

• b) $\int \frac{x^4}{3x^5 + 11} dx$

Background

Topic: Indefinite Integrals (Substitution)

This question tests your ability to use $u$-substitution to evaluate integrals.

Key Steps:

• For (a), let $u = 7x + 2$ and find $du$.

• For (b), let $u = 3x^5 + 11$ and find $du$.

Step-by-Step Guidance

1. Choose an appropriate substitution for each integral.

2. Rewrite the integral in terms of $u$ and $du$.

3. Integrate with respect to $u$.

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Q16. Find the area of the region between the parabola $y = x^2 - 19$ and the line $y = 18x$. The graph is given.

Background

Topic: Area Between Curves

This question tests your ability to set up and evaluate a definite integral representing the area between two curves.

Key Steps:

• Find the points of intersection by setting $x^2 - 19 = 18x$.

• Set up the integral $\int_{a}^{b} [\text{top curve} - \text{bottom curve}] dx$.

Step-by-Step Guidance

1. Solve $x^2 - 19 = 18x$ for $x$ to find the limits of integration.

2. Set up the integral with the correct order (top minus bottom function).

3. Integrate and evaluate at the limits (stop before the final calculation).

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Q17. Find the volume of the solid generated by revolving the region between $y = 49 - x^2$ and $y = 0$ around the x-axis.

Background

Topic: Volumes of Solids of Revolution (Disk/Washer Method)

This question tests your ability to set up and evaluate an integral for the volume of a solid of revolution.

Key Steps:

• Find the $x$-values where $y = 49 - x^2$ meets $y = 0$ (the limits of integration).

• Set up the volume integral using the disk method: $V = \pi \int_{a}^{b} [R(x)]^2 dx$.

Step-by-Step Guidance

1. Solve $49 - x^2 = 0$ for $x$ to find the limits of integration.

2. Set up the integral $V = \pi \int_{a}^{b} (49 - x^2)^2 dx$.

3. Expand and integrate (stop before the final calculation).

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Q18. Let $R$ be the region bounded by $y = 4x - x^2$ and $y = 0$. Use the shell method to find the volume of the solid generated when $R$ is rotated around the y-axis.

Background

Topic: Volumes by Shell Method

This question tests your ability to use the shell method to find the volume of a solid of revolution.

Key Formula:

• Shell method: $V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) dx$

Step-by-Step Guidance

1. Find the $x$-values where $4x - x^2 = 0$ to determine the limits of integration.

2. Set up the shell method integral: $V = 2\pi \int_{a}^{b} x [4x - x^2] dx$.

3. Expand and integrate (stop before the final calculation).

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