Calculus I: Limits, Continuity, and Differentiation Study Guide

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a certain value. Understanding limits is foundational for calculus, especially for defining derivatives and continuity.

• Plug it in: Substitute the value directly into the function if possible.

• Factoring, graphing, tables: Use algebraic manipulation or graphical analysis to evaluate limits.

• Multiply by the conjugate: Useful for limits involving square roots.

• Algebraic/abstract fractions: Simplify complex fractions before evaluating the limit.

• One-sided limits: Limits from the left ($x \to a^-$) and right ($x \to a^+$) may differ. If they are not equal, the two-sided limit does not exist (DNE).

• Limits at infinity: Used to find horizontal asymptotes.

• Limit approaching infinity: Indicates vertical asymptotes.

Continuity

A function is continuous at a point if it is defined there, its limit exists, and the value of the function equals the limit.

• $f(c)$ exists

• $\lim_{x \to c} f(x)$ exists

• $\lim_{x \to c} f(x) = f(c)$

Therefore, $f(x)$ is continuous at $x = c$ if all three conditions are met.

Intermediate Value Theorem (IVT)

• If $f(x)$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists an $x = c$ on $[a, b]$ such that $f(c) = k$.

Introduction to Derivatives

Limit Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point.

• At a point $x$:

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

• Alternate definition at a point $a$:

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

• General function limit:

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

Differentiability

• A function is differentiable at a point if the derivative exists there.

• If a function is differentiable at a point, it is also continuous there, but the converse is not always true.

• Use the limit definition to check left and right hand limits to prove differentiability at a point.

• Identify points where a function is continuous but not differentiable (e.g., sharp corners, cusps, vertical tangents).

Techniques of Differentiation

Basic Derivative Rules

• Sum and Difference Rule

• Constant Multiple Rule

• Product Rule

• Quotient Rule

Find the Derivative Using:

• Polynomial functions

• Trigonometric functions

• Exponential functions

• Logarithmic functions

Second Derivative

• The second derivative $f''(x)$ gives information about the concavity of the function and points of inflection.

Applications of the Derivative

• Slope of a tangent line

• Instantaneous rate of change

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

• Velocity and acceleration: Derivatives of position with respect to time

• Graphical interpretation: Use the graph of $f(x)$ to sketch $f'(x)$ and vice versa

The Chain Rule and Implicit Differentiation

Chain Rule

The chain rule is used to differentiate composite functions.

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

• Can include multiple chains (nested functions).

• Functions given symbolically, graphically, or as a table of values may require the chain rule.

Implicit Differentiation

• Whenever there is a $y$, multiply by $\frac{dy}{dx}$.

• Watch out for the product rule and quotient rule when differentiating implicitly.

Derivatives of Inverse, Exponential, and Logarithmic Functions

Inverse Functions

• Given $g(x) = f^{-1}(x)$, then $g'(a) = \frac{1}{f'(g(a))}$

• Memorize the formula for the derivative of an inverse function.

L'Hôpital's Rule

L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is indeterminate, then:

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

• Show the limit of the numerator is zero.

• Show the limit of the denominator is zero.

• Find the derivatives of the numerator and denominator, then evaluate the limit.

Applications of Derivatives

Extreme Value Theorem (EVT)

• If $f(x)$ is continuous on a closed interval, then $f(x)$ has an absolute maximum and minimum value.

• Find the critical points on the interval.

• Plug the critical points and endpoints into the function to compare their values.

• The maximum is the largest value; the minimum is the smallest value.

Mean Value Theorem (MVT)

• Conditions: $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$.

• Conclusion: There exists $c$ in $(a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}$

• MVT states that the instantaneous rate of change equals the average rate of change at some point.

Relative Extrema and Curve Sketching

• Find critical points by setting $f'(x) = 0$ or where $f'(x)$ is undefined.

• Use the first derivative test to determine if a critical point is a maximum or minimum.

• Second derivative test: If $f''(x) > 0$, the function is concave up (minimum); if $f''(x) < 0$, concave down (maximum).

• Curve sketching: Use the sign of $f'(x)$ and $f''(x)$ to determine intervals of increase/decrease and concavity.

Optimization

• Optimization involves finding the maximum or minimum value of a quantity or expression.

• Draw a picture if applicable.

• Write what is given and what is to be found using variables.

• Write an equation relating the variables.

• Take the derivative of both sides with respect to $t$ (implicit differentiation if necessary).

• Substitute known values and solve for the unknown variable, including appropriate units.

Related Rates

• Related rates problems involve quantities that are changing with respect to time.

• Use implicit differentiation to relate the rates of change of different variables.