Calculus I Midterm Exam Study Guide: Functions, Limits, Differentiation, and Applications

Functions: Domain and Range

Understanding Domain and Range

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce.

• Domain of $g(x)$: For $g(x) = 1 + ext{sqrt}(x+1)$, the domain is $x ext{ such that } x ext{≥} -1$, i.e., $x ext{∈} [-1, ext{∞})$.

• Range of $g(x)$: For $g(x)$ as above, the range is $y ext{≥} 1$, i.e., $y ext{∈} [1, ext{∞})$.

• Domain of $h(x)$: For $h(x) = -e^{-x^2} - 1$, the domain is $x ext{∈} [0, ext{∞})$.

• Range of $h(x)$: For $x > 0$, $-e^{-x^2} < 0$, so $h(x) < -1$. As $x o 0$, $h(x) o -2$; as $x o ext{∞}$, $h(x) o -1$.

Example:

Find the domain and range of $f(x) = ext{sqrt}(x-2)$. Domain: $x ext{≥} 2$. Range: $y ext{≥} 0$.

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a particular value. They are foundational for defining continuity and derivatives.

• Sandwich (Squeeze) Theorem: If $g(x) ext{≤} f(x) ext{≤} h(x)$ and $ext{lim}_{x o a} g(x) = ext{lim}_{x o a} h(x) = L$, then $ext{lim}_{x o a} f(x) = L$.

• One-sided limits: $ext{lim}_{x o a^-} f(x)$ and $ext{lim}_{x o a^+} f(x)$ refer to the limit as $x$ approaches $a$ from the left and right, respectively.

• Continuity: A function $f(x)$ is continuous at $x=a$ if $ext{lim}_{x o a} f(x) = f(a)$.

Example:

Evaluate $ext{lim}_{x o 0} rac{ ext{sin}(2x)}{x}$. Using the standard limit $ext{lim}_{x o 0} rac{ ext{sin}(kx)}{x} = k$, the answer is $2$.

Differentiation: Rules and Applications

Implicit and Logarithmic Differentiation

Implicit differentiation is used when a function is not given explicitly as $y=f(x)$. Logarithmic differentiation is useful for functions involving products, quotients, or powers.

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

• Logarithmic differentiation: Take the natural logarithm of both sides before differentiating.

• Normal and tangent lines: The slope of the tangent line at $(x_0, y_0)$ is $f'(x_0)$. The normal line is perpendicular to the tangent.

Example:

Given $y^2 + x^2 = 1$, differentiate implicitly to find $rac{dy}{dx}$: $2y rac{dy}{dx} + 2x = 0 ightarrow rac{dy}{dx} = - rac{x}{y}$.

Linearization and Approximations

Using Linearization

Linearization approximates a function near a point using its tangent line. For $f(x)$ near $x=a$, the linearization is $L(x) = f(a) + f'(a)(x-a)$.

• Example: Approximate $an^{-1}(1.02)$ using $f(x) = an^{-1}x$ at $a=1$:

$f(1) = rac{ ext{π}}{4}$ $f'(x) = rac{1}{1+x^2}$ $L(x) = rac{ ext{π}}{4} + rac{1}{2}(x-1)$ $L(1.02) = rac{ ext{π}}{4} + rac{0.02}{2} = rac{ ext{π}}{4} + 0.01$

Extreme Values of Functions

Finding Absolute Maximum and Minimum

To find the absolute maximum and minimum of a function on a closed interval, evaluate the function at critical points and endpoints.

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

• Endpoints: Evaluate $f(x)$ at the interval's endpoints.

• Compare values: The largest is the absolute maximum, the smallest is the absolute minimum.

Example:

For $f(x) = ext{ln}(x^2+2x+2)$ on $[1,4]$, find $f'(x)$, set $f'(x)=0$ to find critical points, and evaluate $f(x)$ at $x=1$, $x=4$, and the critical point.

Summary Table: Exam Structure

Points Distribution

The following table summarizes the points assigned to each question in the exam:

Question	Max. Points
1	16
2	20
3	9
4	14
5	12
6	10
7	9
8	10
Total	100

Exam Policies and Instructions

General Guidelines

• The exam duration is 110 minutes and consists of 8 questions over 6 pages.

• Calculators and electronic devices are not allowed.

• Leaving the exam room temporarily is not permitted during the first 35 minutes.

• Answers must be supported by work; unsupported answers receive no credit.

• Cheating or unauthorized communication will result in removal from the exam and reporting to the Dean's office.

Additional info: The study notes above cover the main calculus topics assessed in the midterm exam, including functions, limits, differentiation, linearization, and extreme values, with examples and formulas for each concept.