Calculus I Final Exam Review – Step-by-Step Guidance

Q1. The position of an object moving horizontally is given by $s = f(t) = t^2 - 12t$, for $0 \leq t \leq 13$. Answer the following:

• a) Graph the position function.

• b) Find and graph the velocity function.

• c) When is the object stationary, moving to the right, and moving to the left?

• d) Determine the velocity and acceleration at $t = 1$.

• e) Determine the acceleration when velocity is zero.

• f) On what intervals is the speed increasing?

Background

Topic: Position, Velocity, and Acceleration (Kinematics in Calculus)

This question tests your understanding of how to analyze motion using derivatives: finding velocity and acceleration from a position function, interpreting their meaning, and analyzing intervals of motion.

Key Terms and Formulas

• Position function: $s(t)$

• Velocity: $v(t) = s'(t)$ (first derivative of position)

• Acceleration: $a(t) = v'(t) = s''(t)$ (second derivative of position)

• Speed: $|v(t)|$ (absolute value of velocity)

Step-by-Step Guidance

1. For part (a), plot $s(t) = t^2 - 12t$ for $0 \leq t \leq 13$. Identify key features such as vertex and intercepts.

2. For part (b), find the velocity function by differentiating $s(t)$: $v(t) = \frac{ds}{dt}$.

3. For part (c), set $v(t) = 0$ to find when the object is stationary. Analyze the sign of $v(t)$ to determine when the object moves right (positive velocity) or left (negative velocity).

4. For part (d), evaluate $v(1)$ and $a(1)$ by plugging $t = 1$ into the velocity and acceleration functions.

5. For part (e), solve $v(t) = 0$ for $t$, then substitute this $t$ into $a(t)$ to find the acceleration when velocity is zero.

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Q2. Evaluate the following limits:

• a) $\lim_{x \to -\infty} x^3$

• b) $\lim_{x \to 3} \frac{x-3}{x^2-3}$

• c) $\lim_{x \to \infty} \frac{2x-3}{4x+10}$

• d) $\lim_{x \to \infty} \frac{x-7}{x(x-5)^2}$

Background

Topic: Limits and Asymptotic Behavior

This question tests your ability to evaluate limits, including at infinity and at points where the function may be undefined.

Key Terms and Formulas

• Limit: $\lim_{x \to a} f(x)$

• For rational functions as $x \to \infty$, compare degrees of numerator and denominator.

• Indeterminate forms may require algebraic manipulation.

Step-by-Step Guidance

1. For (a), consider the behavior of $x^3$ as $x$ becomes very large negative.

2. For (b), substitute $x = 3$ and check if the limit is defined or if you get an indeterminate form. If so, factor or simplify.

3. For (c), divide numerator and denominator by $x$ to analyze the limit as $x \to \infty$.

4. For (d), expand the denominator and compare the highest degree terms in numerator and denominator.

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Q3. Use the definition of the derivative to find the slope of the tangent line to $f(x) = \frac{1}{2x+1}$ at $P(0,1)$. Then, determine the equation of the tangent line at $P$.

Background

Topic: Derivative Definition and Tangent Lines

This question tests your ability to use the limit definition of the derivative to find the slope at a point and write the equation of the tangent line.

Key Terms and Formulas

• Definition of derivative: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Equation of tangent line: $y - f(a) = f'(a)(x - a)$

Step-by-Step Guidance

1. Write $f(x) = \frac{1}{2x+1}$ and identify $a = 0$.

2. Compute $f(0)$ and $f(0+h)$.

3. Set up the difference quotient: $\frac{f(0+h) - f(0)}{h}$.

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

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Q4. Find the derivative of the following functions:

• a) $y = \tan(\sin \theta)$

• b) $y = \frac{\ln x}{\ln x + 1}$

• c) $f(x) = x e^{-10x}$

• d) $f(x) = e^{6x} \sin x$

Background

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

This question tests your ability to apply various differentiation rules to composite, quotient, and product functions.

Key Terms and Formulas

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

• Quotient Rule: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$

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

• Derivatives of basic functions: $\frac{d}{dx} \ln x = \frac{1}{x}$, $\frac{d}{dx} e^{ax} = a e^{ax}$, $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \tan x = \sec^2 x$

Step-by-Step Guidance

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

2. Write the derivative formula for the function, substituting the appropriate inner and outer functions.

3. For composite functions, apply the chain rule carefully, differentiating the outer function and multiplying by the derivative of the inner function.

4. For products and quotients, apply the product or quotient rule as needed, keeping track of each term.

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Q5. Find $\frac{dy}{dx}$ for the curve $x + y = \cos(2x)$.

Background

Topic: Implicit Differentiation

This question tests your ability to use implicit differentiation to find $\frac{dy}{dx}$ when $y$ is not isolated.

Key Terms and Formulas

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

• Chain rule: $\frac{d}{dx} y = \frac{dy}{dx}$

• Derivative of $\cos(2x)$: $-2 \sin(2x)$

Step-by-Step Guidance

1. Differentiate both sides of the equation $x + y = \cos(2x)$ with respect to $x$.

2. Remember to use $\frac{dy}{dx}$ when differentiating $y$.

3. Solve the resulting equation for $\frac{dy}{dx}$.

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Q6. Find the critical points for $f(x) = 2x^5 - 5x^4 - 10x^3 + 4$ on $[-2,4]$. Then determine the points for the absolute minimum and maximum values of $f$ on the interval.

Background

Topic: Critical Points and Absolute Extrema

This question tests your ability to find critical points by setting the derivative to zero, and to determine absolute extrema on a closed interval.

Key Terms and Formulas

• Critical points: Where $f'(x) = 0$ or $f'(x)$ is undefined.

• Absolute extrema: Evaluate $f$ at critical points and endpoints.

Step-by-Step Guidance

1. Find $f'(x)$ and set it equal to zero to solve for critical points in $[-2,4]$.

2. Check if $f'(x)$ is undefined anywhere in the interval.

3. Evaluate $f(x)$ at all critical points and at the endpoints $x = -2$ and $x = 4$.

4. Compare these values to determine which are absolute minimum and maximum.

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Q7. Calculate the right Riemann sum for $f(x) = x^3 - 1$ on $[-2,0]$ using $n = 4$.

Background

Topic: Riemann Sums and Approximating Area

This question tests your ability to approximate the area under a curve using a right Riemann sum.

Key Terms and Formulas

• Right Riemann sum: $\sum_{i=1}^n f(x_i) \Delta x$ where $x_i$ are right endpoints.

• $\Delta x = \frac{b-a}{n}$

Step-by-Step Guidance

1. Calculate $\Delta x$ for the interval $[-2,0]$ with $n=4$.

2. Find the right endpoints $x_i$ for $i=1$ to $4$.

3. Evaluate $f(x_i)$ at each right endpoint.

4. Set up the sum $\sum_{i=1}^4 f(x_i) \Delta x$.

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Q8. Evaluate the following definite integrals (exact answers):

• a) $\int_0^2 4x^3 \, dx$

• b) $\int_0^2 (3x^2 + 2x) \, dx$

• c) $\int_0^{\ln 8} e^x \, dx$

• d) $\int_0^{\frac{\pi}{4}} \sec^2 x \, dx$

Background

Topic: Definite Integrals and Fundamental Theorem of Calculus

This question tests your ability to compute definite integrals using antiderivatives and evaluate them at the bounds.

Key Terms and Formulas

• Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$ where $F$ is an antiderivative of $f$.

• Common antiderivatives: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $\int e^x dx = e^x + C$, $\int \sec^2 x dx = \tan x + C$

Step-by-Step Guidance

1. Find the antiderivative for each integrand.

2. Evaluate the antiderivative at the upper and lower bounds.

3. Subtract the lower value from the upper value for each integral.

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Q9. Use substitution to solve the indefinite integral $\int x^9 \sin(x^{10}) dx$.

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to simplify and solve an integral.

Key Terms and Formulas

• Let $u = x^{10}$, then $du = 10x^9 dx$

• Rewrite the integral in terms of $u$ and $du$

Step-by-Step Guidance

1. Let $u = x^{10}$ and compute $du$ in terms of $dx$.

2. Rewrite $x^9 dx$ in terms of $du$.

3. Substitute into the original integral to express everything in terms of $u$.

4. Integrate with respect to $u$.

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Q10. Does the Mean Value Theorem apply to $f(x) = 7 - x^2$ on $[-1,2]$? If so, find the point(s) guaranteed to exist by the theorem.

Background

Topic: Mean Value Theorem (MVT) for Derivatives

This question tests your understanding of the conditions for the MVT and how to find the value(s) $c$ that satisfy the theorem.

Key Terms and Formulas

• MVT: If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

Step-by-Step Guidance

1. Check if $f(x)$ is continuous and differentiable on the given interval.

2. Compute $f(-1)$ and $f(2)$.

3. Calculate the average rate of change $\frac{f(2) - f(-1)}{2 - (-1)}$.

4. Set $f'(c)$ equal to this value and solve for $c$ in $(-1,2)$.

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Q11. Use l'Hôpital's Rule to evaluate $\lim_{x \to 3} \frac{\ln x}{4x - x^2 - 3}$.

Background

Topic: L'Hôpital's Rule for Indeterminate Forms

This question tests your ability to recognize indeterminate forms and apply l'Hôpital's Rule to evaluate limits.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Substitute $x = 3$ into the numerator and denominator to check for an indeterminate form.

2. If indeterminate, differentiate numerator and denominator separately.

3. Take the limit of the new fraction as $x \to 3$.

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Q12. Given $v(0) = 70$, $s(0) = 10$, and $a(t) = -32$, find the equation for the position function $s(t)$.

Background

Topic: Antiderivatives and Initial Value Problems (Kinematics)

This question tests your ability to find a position function from acceleration, given initial velocity and position.

Key Terms and Formulas

• Acceleration: $a(t) = s''(t)$

• Integrate $a(t)$ to get $v(t)$, then integrate $v(t)$ to get $s(t)$

• Use initial conditions to solve for constants of integration

Step-by-Step Guidance

1. Integrate $a(t) = -32$ to find $v(t)$, introducing a constant $C_1$.

2. Use $v(0) = 70$ to solve for $C_1$.

3. Integrate $v(t)$ to find $s(t)$, introducing a constant $C_2$.

4. Use $s(0) = 10$ to solve for $C_2$.

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