Calculus I Midterm Study Guide: Limits, Continuity, Derivatives, and Applications

Limits and Continuity

Evaluating Limits

Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Calculating limits often involves algebraic manipulation, application of the Squeeze Theorem, or recognizing indeterminate forms.

• Absolute Value and Oscillating Functions: For expressions like $\lim_{x \to 0} |x| \cos\left(\frac{1}{x}\right)$, the Squeeze Theorem is useful since $|\cos\left(\frac{1}{x}\right)| \leq 1$ and $|x|$ approaches 0.

• Limits at Infinity: For rational expressions such as $\lim_{x \to \infty} \frac{\sqrt{x^2 + 3x + 1}}{3x}$, divide numerator and denominator by $x$ to simplify.

• One-Sided Limits: For $\lim_{x \to 5^-} \frac{2}{x-5}$, consider the sign and behavior as $x$ approaches 5 from the left.

Types of Discontinuities

Discontinuities occur where a function is not continuous. They are classified as follows:

• Removable Discontinuity: The limit exists, but the function is not defined or is defined differently at that point.

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite Discontinuity: The function approaches infinity at the discontinuity.

Intervals of Continuity

A function is continuous on an interval if it has no discontinuities within that interval.

• Identify intervals by examining the graph and noting where the function is unbroken.

Limit Evaluation from Graphs

Limits can be estimated from graphs by observing the behavior of the function as $x$ approaches a given value from the left and right.

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

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

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

Derivatives

Limit Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change. The formal definition is:

• $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Apply this definition to $f(x) = 3x^2$ to find $f'(x)$ algebraically.

Power Rule for Derivatives

The Power Rule is a shortcut for differentiating functions of the form $x^n$:

• $\frac{d}{dx} x^n = n x^{n-1}$

• For $f(x) = 3x^2$, $f'(x) = 6x$

Evaluating Derivatives at a Point

To find the slope of the tangent line at a specific point, substitute the value into the derivative:

• For $x = 1$, $f'(1) = 6 \times 1 = 6$

Equation of the Tangent Line

The tangent line to $y = f(x)$ at $x = a$ is given by:

• $y = f'(a)(x - a) + f(a)$

• For $f(x) = 3x^2$ at $x = 1$, $y = 6(x - 1) + 3$

Applications of Derivatives

Velocity: Average and Instantaneous

For a position function $s(t)$, velocity is the rate of change of position:

• Average velocity: $\frac{s(b) - s(a)}{b - a}$ over $[a, b]$

• Instantaneous velocity: $v(t) = s'(t)$

Finding Instantaneous Velocity

Differentiate the position function to obtain velocity:

• For $s(t) = \frac{3t - 1}{t + 1}$, use the quotient rule:

• $v(t) = \frac{(t + 1) \cdot 3 - (3t - 1) \cdot 1}{(t + 1)^2}$

Solving for Zero Position

To find when $s(t) = 0$, solve $3t - 1 = 0 \Rightarrow t = \frac{1}{3}$, and use the Intermediate Value Theorem to justify the existence of a solution in an interval.

Key Calculus Formulas and Theorems

Basic Derivative Rules

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

• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

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

Trigonometric Derivatives

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} \tan x = \sec^2 x$

Mean Value Theorem (MVT)

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that:

• $f'(c) = \frac{f(b) - f(a)}{b - a}$

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c$ in $(a, b)$ such that $f(c) = N$.

Definite Integral Formula

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