Math 111 Final Exam Calculus Study Guidance

Q1. Evaluate the following limits:

• (a) $\lim_{x \to 0} \frac{\sin x^2}{\cos 2x - 1}$

• (b) $\lim_{x \to 0} (e^{2x} + x)^{1/x}$

• (c) $\lim_{t \to -2} \frac{t^3 - 2t + 4}{t^2 - t - 6}$

Background

Topic: Limits and L'Hospital's Rule

These problems test your understanding of evaluating limits, including indeterminate forms and algebraic manipulation. You may need to use L'Hospital's Rule, Taylor expansions, or algebraic simplification.

Key Terms and Formulas:

• Limit: $\lim_{x \to a} f(x)$

• L'Hospital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is $0/0$ or $\infty/\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit exists).

• Taylor expansion for small $x$: $\sin x \approx x$, $e^x \approx 1 + x$

Step-by-Step Guidance

1. For each limit, first check if direct substitution gives an indeterminate form (like $0/0$ or $1^\infty$).

2. If indeterminate, consider algebraic simplification, Taylor expansion, or L'Hospital's Rule as appropriate.

3. For (a), expand $\sin x^2$ and $\cos 2x$ for small $x$ using Taylor series.

4. For (b), rewrite the expression using logarithms to handle the $1/x$ exponent, and consider the limit of the exponent.

5. For (c), factor numerator and denominator to see if you can cancel terms before substituting $t = -2$.

Try solving on your own before revealing the answer!

Q2. Find $\frac{dy}{dx}$ for each of the following:

• (a) $y = e^{x+y} - x$

• (b) $y = x \arcsin(2x)$

• (c) $y = x^{1 + \frac{1}{\ln x}}$

Background

Topic: Differentiation (Implicit, Product, Chain, and Logarithmic Differentiation)

These problems test your ability to differentiate functions, including implicit differentiation and using the product, chain, and logarithmic rules.

Key Terms and Formulas:

• Implicit differentiation: Differentiate both sides with respect to $x$, treating $y$ as a function of $x$.

• Product rule: $\frac{d}{dx}[uv] = u'v + uv'$

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

• Logarithmic differentiation: Take $\ln$ of both sides to simplify exponents.

Step-by-Step Guidance

1. For (a), differentiate both sides with respect to $x$, remembering to use the chain rule for $e^{x+y}$ and implicit differentiation for $y$.

2. For (b), apply the product rule to $x \arcsin(2x)$, and use the chain rule for $\arcsin(2x)$.

3. For (c), use logarithmic differentiation: take $\ln$ of both sides, differentiate, and then solve for $\frac{dy}{dx}$.

4. After differentiating, isolate $\frac{dy}{dx}$ where necessary.

Try solving on your own before revealing the answer!

Q3. Evaluate the integrals:

• (a) $\int_{-1}^1 \frac{x}{1 + x^2} dx$

• (b) $\int 3x^2 \sqrt{5x^3 + 9} dx$

• (c) $\int 3x^2(1 - x^{-6}) dx$

Background

Topic: Integration (Substitution, Definite and Indefinite Integrals)

These problems test your ability to evaluate definite and indefinite integrals, including substitution and handling powers and roots.

Key Terms and Formulas:

• Substitution: Let $u = g(x)$, then $du = g'(x)dx$

• Power rule for integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)

• Definite integral: $\int_a^b f(x) dx = F(b) - F(a)$ where $F$ is an antiderivative of $f$

Step-by-Step Guidance

1. For (a), consider substitution $u = 1 + x^2$ and check the symmetry of the integrand over $[-1, 1]$.

2. For (b), let $u = 5x^3 + 9$ and find $du$ in terms of $dx$.

3. For (c), expand the integrand and integrate each term separately using the power rule.

4. For definite integrals, remember to evaluate the antiderivative at the upper and lower limits.

Try solving on your own before revealing the answer!

Q4. Applications of Integration and Optimization

• (a) Find the area of the region bounded by $y = x^2 - 6$ and $y = 2x + 2$.

• (b) A rectangular plot of land will be bounded on one side by a stream and the other three sides by a fence. If $800$ft$^2$ of land is to be enclosed, what are the dimensions that will require the least amount of fencing? Show that your result is a minimum.

Background

Topic: Area Between Curves and Optimization

These problems test your ability to set up and solve area problems using definite integrals and to solve optimization problems using calculus.

Key Terms and Formulas:

• Area between curves: $A = \int_a^b [f(x) - g(x)] dx$ where $f(x) \geq g(x)$ on $[a, b]$

• Optimization: Use constraints to write the quantity to be minimized/maximized as a function of one variable, then use calculus to find critical points.

Step-by-Step Guidance

1. For (a), set $x^2 - 6 = 2x + 2$ to find intersection points (limits of integration).

2. Set up the integral for the area between the curves using the correct upper and lower functions.

3. For (b), let $x$ and $y$ be the dimensions of the rectangle. Write the area constraint: $xy = 800$.

4. Express the amount of fencing needed in terms of $x$ and $y$, then use the constraint to write it as a function of one variable.

5. Take the derivative, set it to zero to find critical points, and use the second derivative test to confirm a minimum.

Try solving on your own before revealing the answer!

Q5. Find $\frac{dy}{dx}$ for each of the following:

• (a) $y = \ln(\cos e^x)$

• (b) $y = x \tan(2\sqrt{x}) + 4$

• (c) $y = \int_0^{\sqrt{x}} \cos t^2 dt$

Background

Topic: Differentiation (Chain Rule, Product Rule, Fundamental Theorem of Calculus)

These problems test your ability to differentiate composite functions and apply the Fundamental Theorem of Calculus for variable upper limits.

Key Terms and Formulas:

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

• Product rule: $\frac{d}{dx}[uv] = u'v + uv'$

• Fundamental Theorem of Calculus: $\frac{d}{dx} \int_a^{g(x)} f(t) dt = f(g(x))g'(x)$

Step-by-Step Guidance

1. For (a), use the chain rule to differentiate $\ln(\cos e^x)$.

2. For (b), apply the product rule to $x \tan(2\sqrt{x})$ and use the chain rule for $\tan(2\sqrt{x})$.

3. For (c), apply the Fundamental Theorem of Calculus with a variable upper limit $\sqrt{x}$.

4. Remember to multiply by the derivative of the upper limit when applying the theorem.

Try solving on your own before revealing the answer!

Q6. Applications of Integration and Related Rates

• (a) Find the average value of the function $f(t) = 3t^2 - 2t - 4$ over the interval $[1, 3]$.

• (b) Air is being pumped into a spherical balloon at a rate of $20$in$^3$/s. How fast is the radius increasing when the surface area is $4$ in$^2$? (Recall, $V = \frac{4}{3}\pi r^3$ and $A = 4\pi r^2$).

Background

Topic: Average Value of a Function and Related Rates

These problems test your ability to compute the average value of a function using integrals and to solve related rates problems involving geometric formulas.

Key Terms and Formulas:

• Average value: $f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$

• Related rates: Differentiate both sides of a geometric formula with respect to time $t$.

• For a sphere: $V = \frac{4}{3}\pi r^3$, $A = 4\pi r^2$

Step-by-Step Guidance

1. For (a), set up the average value formula and compute the definite integral of $f(t)$ from $1$to $3$.

2. Divide the result by the length of the interval $(3-1)$.

3. For (b), write $\frac{dV}{dt} = 20$ and relate $\frac{dr}{dt}$ to $\frac{dV}{dt}$ using the derivative of the volume formula.

4. Use the given surface area to solve for $r$ at the instant in question.

5. Substitute $r$ into your differentiated equation to solve for $\frac{dr}{dt}$.

Try solving on your own before revealing the answer!

Q7. Consider the function $y = \frac{4 + x^2}{x}$

• (a) Find the intervals on which this function is increasing or decreasing

• (b) Find the intervals on which this function is concave up or concave down

• (c) Find all asymptotes; horizontal, vertical and/or slant.

• (d) Determine the points (if any) at which this function has a local maximum, a local minimum or a point of inflection

• (e) Sketch this function making sure to label the points found in part d.

Background

Topic: Curve Sketching (First and Second Derivative Tests, Asymptotes)

This problem tests your ability to analyze a rational function using derivatives to determine intervals of increase/decrease, concavity, and to find asymptotes and critical points.

Key Terms and Formulas:

• First derivative: $y' = \frac{d}{dx} y$ (for increasing/decreasing)

• Second derivative: $y'' = \frac{d^2}{dx^2} y$ (for concavity)

• Vertical asymptote: where denominator is zero

• Horizontal/slant asymptote: analyze end behavior as $x \to \infty$ or $x \to -\infty$

Step-by-Step Guidance

1. Rewrite $y = \frac{4 + x^2}{x}$ as $y = \frac{4}{x} + x$ to simplify differentiation.

2. Compute the first derivative $y'$ and set it to zero to find critical points (for increasing/decreasing).

3. Compute the second derivative $y''$ and set it to zero to find possible inflection points (for concavity).

4. Find where the denominator is zero for vertical asymptotes, and analyze limits as $x \to \infty$ for horizontal/slant asymptotes.

5. Use the first and second derivative tests to classify critical points and inflection points.

Try solving on your own before revealing the answer!