Calculus I: Derivatives, Tangents, and Continuity – Unit 3 Review Study Guide

Review for Unit 3 Test

Implicit Differentiation and Tangent Lines

Implicit differentiation is used when a function is defined implicitly by an equation involving both x and y. This technique allows us to find the derivative dy/dx even when y is not isolated.

• Implicit Differentiation: Differentiate both sides of the equation with respect to x, treating y as a function of x (using the chain rule).

• Example: For the curve defined by $x^2 + xy + y^2 = 27$:

  • Differentiating both sides: $2x + y + x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$

  • Solving for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{-2x - y}{x + 2y}$

• Vertical Tangents: Occur where the denominator of $\frac{dy}{dx}$ is zero (i.e., $x + 2y = 0$).

• Horizontal Tangents: Occur where the numerator of $\frac{dy}{dx}$ is zero (i.e., $-2x - y = 0$).

• Equation of Tangent Line: At point $(x_0, y_0)$, the tangent line is $y - y_0 = m(x - x_0)$, where $m = \frac{dy}{dx}$ at $(x_0, y_0)$.

• Rate of Change of Slope: The second derivative $\frac{d^2y}{dx^2}$ gives the rate of change of the slope of the curve.

Derivatives of Explicit Functions

Finding derivatives of explicit functions involves applying differentiation rules such as the power rule, product rule, quotient rule, and chain rule.

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

• Examples:

  • $f(x) = \sin(e^x)$: $f'(x) = \cos(e^x) \cdot e^x$

  • $g(x) = \ln(x^2 + 1)$: $g'(x) = \frac{2x}{x^2 + 1}$

  • $h(x) = x^5 e^{x^2}$: $h'(x) = 5x^4 e^{x^2} + x^5 \cdot 2x e^{x^2}$

Derivatives from Tabular Data

When functions are defined by tables, derivatives can be computed using the values provided and differentiation rules.

x	f(x)	g(x)
1	2	2
2	3	1
3	1/2	4
4	3	-1

• Example: If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$, using the table values for $f(x)$, $g(x)$, $f'(x)$, and $g'(x)$.

Inverse Functions and Their Derivatives

The derivative of an inverse function at a point can be found using the formula:

• $\left(f^{-1}\right)'(a) = \frac{1}{f'(f^{-1}(a))}$

• Example: If $f(x) = \sin x$, then $f^{-1}(x) = \arcsin x$ and $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$

Limits and Continuity

Limits are foundational to calculus and are used to define derivatives and continuity. A function is continuous at a point if the limit from both sides equals the function value at that point.

• Limit Laws: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• Types of Discontinuity:

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

  • Infinite Discontinuity: Function approaches infinity at a point.

  • Removable Discontinuity: Limit exists but function is not defined at that point.

• Example: $f(x) = \frac{3x^2 + 5x}{x^2 - 2x}$ is not continuous at $x = 0$ and $x = 2$ (denominator zero).

Differentiability

A function is differentiable at a point if it is continuous there and its derivative exists. Points of non-differentiability include corners, cusps, and discontinuities.

• Example: $f(x) = |x|$ is not differentiable at $x = 0$ because the left and right derivatives are not equal.

Graphical Analysis of Derivatives

Given graphs of $f(x)$ and $g(x)$, derivatives can be estimated by analyzing slopes of tangent lines or using piecewise definitions.

• Example: If $f(x)$ is linear between points, $f'(x)$ is the slope of the segment.

Summary Table: Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx}[u v] = u' v + u v'$	$\frac{d}{dx} (x^2 e^x) = 2x e^x + x^2 e^x$
Quotient Rule	$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$	$\frac{d}{dx} \left(\frac{x}{x+1}\right) = \frac{1(x+1) - x(1)}{(x+1)^2}$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$	$\frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x$

Additional info:

• Some problems involve finding derivatives at specific points, using both algebraic and graphical methods.

• Inverse trigonometric and logarithmic differentiation are included.

• Continuity and differentiability are tested with piecewise and absolute value functions.