Calculus Study Guide: Tangent Lines, Derivatives, and Applications

Tangent Lines and Linearization

Definition and Applications

The tangent line to a function at a point provides the best linear approximation to the function near that point. Linearization uses the tangent line to estimate function values close to the point of tangency.

• Tangent Line Equation: For a differentiable function f(x) at x = a, the tangent line at (a, f(a)) is given by: $y = f(a) + f'(a)(x - a)$

• Linearization: The linear approximation of f(x) near x = a is: $L(x) = f(a) + f'(a)(x - a)$

• Example: If T(y) represents temperature and T(2020) = 15.01, T'(2020) = 0.03, the tangent line at y = 2020 is: $L(y) = 15.01 + 0.03(y - 2020)$

Graphical Interpretation

• Tangent Line on a Graph: The tangent line touches the curve at one point and has the same slope as the curve at that point.

• Estimating Values: Use the tangent line to approximate values of the function near the point of tangency.

Derivatives and Their Interpretation

Definition and Notation

The derivative of a function measures the rate of change of the function with respect to its variable. It is denoted as f'(x) or dy/dx.

• Derivative Table: A table may list values of f'(x) at specific points. For example:

x	0	1	2	3	4
f'(x)	1	2	3	2	1

• Example: Given f(2) = 5 and f'(2) = 3, the tangent line at x = 2 is: $y = 5 + 3(x - 2)$

Sketching Functions from Derivative Information

• Increasing/Decreasing: If f'(x) > 0, f(x) is increasing; if f'(x) < 0, f(x) is decreasing.

• Concavity: If f''(x) > 0, f(x) is concave up; if f''(x) < 0, f(x) is concave down.

Analyzing Graphs of Functions and Their Derivatives

Critical Points and Extrema

Critical points occur where f'(x) = 0 or f'(x) is undefined. These points may correspond to local maxima, minima, or points of inflection.

• Local Maximum: If f'(x) changes from positive to negative at x = c, f(x) has a local maximum at c.

• Local Minimum: If f'(x) changes from negative to positive at x = c, f(x) has a local minimum at c.

• Inflection Point: Where f''(x) changes sign, the function has an inflection point.

Average Rate of Change

• Formula: The average rate of change of f(x) from x = a to x = b is: $\frac{f(b) - f(a)}{b - a}$

Ordering Derivative Values

• Given several derivative values, you may be asked to order them by magnitude or sign.

Limits and Continuity

Definition of Limit

The limit of f(x) as x approaches a describes the value that f(x) gets close to as x gets close to a.

• Limit Notation: $\lim_{x \to a} f(x)$

• Continuity: f(x) is continuous at x = a if $\lim_{x \to a} f(x) = f(a)$.

• Differentiability: f(x) is differentiable at x = a if the derivative exists at a.

Derivative Rules and Computation

Product and Quotient Rules

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

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

Chain Rule

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

Examples of Derivative Computation

• Find the derivative of: $f(x) = (x^2 - 3x)\sin(x)$

• Find the derivative of: $g(x) = \frac{1}{x^2 - 3x + 1}$

• Find the derivative of: $h(x) = e^{\cot(x)}$

Higher Order Derivatives

Definition

The nth derivative of a function is the derivative of its (n-1)th derivative. For example, the fourth derivative is:

• $f^{(4)}(x) = \frac{d^4}{dx^4}f(x)$

Applications: Optimization and Motion

Optimization Problems

• Finding Maximum/Minimum: Use critical points and endpoints to find absolute extrema on a closed interval.

• Example: For a box with fixed volume, minimize surface area by setting up equations and using derivatives.

Motion Problems

• Velocity: The derivative of position with respect to time.

• Acceleration: The derivative of velocity with respect to time.

• Example: If $s(t) = -t^4 + 18t^2 - 24t$, then: Velocity: $v(t) = s'(t)$ Acceleration: $a(t) = s''(t)$

Graphical Analysis of Differentiability and Extrema

Non-Differentiable Points

• Definition: A function is not differentiable at points where it has a sharp corner, cusp, or vertical tangent.

Local Maxima and Minima from Graphs

• Identify where the function changes direction to find local extrema.

Summary Table: Derivative Properties

Property	Description
Increasing	f'(x) > 0
Decreasing	f'(x) < 0
Concave Up	f''(x) > 0
Concave Down	f''(x) < 0
Critical Point	f'(x) = 0 or undefined
Inflection Point	f''(x) changes sign

Additional info:

• Some problems require sketching graphs based on derivative tables or analyzing the sign of derivatives to determine intervals of increase/decrease and concavity.

• Optimization problems involve setting up equations for area, volume, or other quantities and using derivatives to find extrema.

• Motion problems use derivatives to find velocity and acceleration from position functions.