Visual Inspection Problems

Slopes at Specific Points on a Graph

Understanding the slope of a function at a given point is fundamental in calculus. The slope represents the rate of change of the function at that point, and can be classified as positive, negative, or zero, with varying magnitudes.

• Positive Slope: The function is increasing at the point.

• Negative Slope: The function is decreasing at the point.

• Zero Slope: The function is flat (horizontal tangent) at the point.

• Large Magnitude: Steep slope (rapid change).

• Small Magnitude: Gentle slope (slow change).

Example: Given labeled points on a curve, classify the slope at each as large negative, small negative, zero, small positive, or large positive.

Limits from Graphs

Limits describe the behavior of a function as the input approaches a specific value. Visual inspection of a graph can help determine left-hand and right-hand limits, whether a limit exists, and the value of the function at a point.

• Left-hand limit:

• Right-hand limit:

• Limit at a point:

• Function value:

Example: For a given graph, determine the above limits and function value at .

Limits at Infinity

Limits as approaches positive or negative infinity describe the end behavior of a function. These are important for understanding asymptotes and long-term trends.

• Limit as :

• Limit as :

Example: Identify the indicated limits for a graph as approaches positive and negative infinity.

Continuity, Differentiability, and Definedness

These properties describe the behavior of functions at specific points:

• Continuous at : No breaks, jumps, or holes at .

• Differentiable at : The function has a well-defined tangent (no sharp corners or cusps) at .

• Defined at : The function has a value at .

Example: For a given graph, indicate YES/NO for continuity, differentiability, and definedness at specified -values.

Problems to Work Out

Equation of a Tangent Line

The tangent line to a curve at a specific point represents the instantaneous rate of change. To find its equation:

1. Find the derivative .

2. Evaluate to get the slope at .

3. Find for the point of tangency.

4. Use the point-slope form: .

Example: For , find the tangent at .

Limits of Algebraic Expressions

Limits can be evaluated for polynomials and rational expressions as approaches specific values or infinity.

• Direct Substitution: For polynomials, substitute the value directly.

• Rational Expressions: Simplify and check for indeterminate forms.

• Limits at Infinity: Analyze degrees of numerator and denominator.

Example:

Limit Definition of the Derivative

The derivative of a function at a point is defined as:

Example: Use the limit definition to find the derivative of .

Rules for Finding Derivatives

Several rules simplify the process of differentiation:

• Addition/Subtraction Rule:

• Multiplication Rule (Product Rule):

• Constant Rule:

• Power Rule:

• General Power Rule:

Examples:

• Find for

• Find for

• Find for

• Find for

• Find for

• Find for

Partial Derivatives (Functions of Multiple Variables)

When a function depends on more than one variable, partial derivatives measure the rate of change with respect to one variable, holding others constant.

• Partial derivative with respect to :

• Partial derivative with respect to :

Examples:

• For , find

• For , find

Higher-Order Derivatives

Higher-order derivatives describe the rate of change of previous derivatives. The second derivative measures concavity, and the third derivative measures the rate of change of concavity.

• Second derivative:

• Third derivative:

Examples:

• For , find

• For , find

Evaluating Derivatives at Specific Values

To evaluate a derivative at a specific value, substitute the value into the derivative function.

• Example: For , find

Marginal Cost, Marginal Profit, and Marginal Revenue

In business calculus, derivatives are used to find marginal cost, profit, and revenue, which represent the instantaneous rate of change with respect to production.

• Marginal Cost: , the derivative of the cost function.

• Marginal Profit: , the derivative of the profit function.

• Marginal Revenue: , the derivative of the revenue function.

Example: For ,

• Find the marginal cost equation:

• Find marginal cost for an extra two units produced

• Interpret the result: Marginal cost estimates the cost of producing one additional unit at a given production level.