Calculus I: Limits, Derivatives, and Applications – Study Notes

Limits and Their Evaluation

Definition and Techniques

Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Evaluating limits is essential for understanding continuity, derivatives, and integrals.

• Limit Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) approaches as x gets arbitrarily close to a.

• Notation: $\lim_{x \to a} f(x)$

• Techniques:

  • Direct substitution

  • Factoring and simplifying

  • Rationalizing numerator or denominator

  • L'Hôpital's Rule for indeterminate forms

• Example: $\lim_{x \to \frac{\pi}{2}} \frac{\cos x}{\pi - 2x}$

• Example: $\lim_{x \to 0} \frac{e^x - 1}{x}$

Differentiation: Finding Derivatives

Basic Rules and Applications

Differentiation is the process of finding the derivative of a function, which represents the rate of change or slope of the function at any point.

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

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

• Trigonometric Derivatives:

  • $\frac{d}{dx} \sin x = \cos x$

  • $\frac{d}{dx} \cos x = -\sin x$

  • $\frac{d}{dx} \tan x = \sec^2 x$

• Inverse Trigonometric Derivatives:

  • $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$

  • $\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$

• Logarithmic and Exponential Derivatives:

  • $\frac{d}{dx} \ln x = \frac{1}{x}$

  • $\frac{d}{dx} e^{f(x)} = e^{f(x)} f'(x)$

• Example: $y = \tan(\cos x + \sqrt{x})$ (Apply chain rule and derivatives of tan, cos, and sqrt)

• Example: $y = \cos^2 x + \sin^2 x$ (Differentiate each term and use trigonometric identities)

• Example: $x^2 + y^2 = 9$ (Implicit differentiation)

• Example: $f(x) = \frac{1}{\sin^2 x}$ (Rewrite as $\sin^{-2} x$ and use chain rule)

• Example: $f(x) = e^{x^2 + x} \cos(x^2)$ (Product and chain rules)

• Example: $g(x) = (x^2 + \sin x)^4$ (Chain rule)

• Example: $y = \ln\left(\frac{x^2 - x}{x^2 + 4}\right)$ (Logarithmic differentiation)

• Example: $y = 7 \log_5(x + 4)$ (Change of base and derivative of log)

• Example: $y = 3x^{1/x}$ (Logarithmic differentiation)

• Example: $4 \cos x \sin y = 1$ (Implicit differentiation)

• Example: $y = \sin^{-1}(\cos x)$ (Chain rule and inverse trig derivative)

Related Rates

Solving Problems Involving Rates of Change

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using implicit differentiation and the chain rule.

• General Approach:

  1. Identify all variables and their rates of change.

  2. Write an equation relating the variables.

  3. Differentiating both sides with respect to time (t).

  4. Substitute known values and solve for the desired rate.

• Example: In an electric circuit, $V = RI$. If $V$ is decreasing at 1 volt/sec and $I$ is decreasing at 2 amps/sec, find the rate at which $R$ is changing when $V = 12$ volts and $I = 2$ amps.

Implicit Differentiation

Finding Derivatives When y is Not Explicitly Solved

Implicit differentiation is used when a function is defined implicitly rather than explicitly. This technique is essential for finding derivatives of equations involving both x and y.

• Process:

  1. Differentiate both sides of the equation with respect to x.

  2. Treat y as a function of x (use chain rule: $\frac{d}{dx} y = \frac{dy}{dx}$).

  3. Solve for $\frac{dy}{dx}$.

• Example: $x^2 + y^2 = 9$

• Example: $4 \cos x \sin y = 1$

Derivatives of Inverse Trigonometric Functions

Key Formulas and Applications

Inverse trigonometric functions have specific derivatives that are useful in calculus, especially in integration and solving equations.

• Key Formulas:

  • $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$

  • $\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$

  • Example: Show that $\frac{d}{dx} (\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}$ for $0 \leq y \leq \pi$

Equations of Tangent Lines

Finding the Tangent Line to a Curve at a Point

The equation of the tangent line to a curve at a given point can be found using the derivative (slope) at that point and the point-slope form of a line.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $m$ is the slope at $(x_1, y_1)$

• Example: For the circle $x^2 + y^2 = 100$, find $\frac{dy}{dx}$ and the tangent line at points (6,8) and (10,0).

Linear Approximation and Differentials

Estimating Function Values Near a Point

Linear approximation uses the tangent line to estimate the value of a function near a given point. Differentials provide a way to approximate small changes in function values.

• Linear Approximation Formula: $f(x) \approx f(a) + f'(a)(x - a)$

• Example: Approximate $\sqrt{1001}$ using linear approximation at $x = 1000$.