Calculus: Curve Sketching

Curve Sketching Overview

• Chapter uses the derivative to sketch curves by determining where the curve slopes upwards or downwards, and concavity.
• This chapter extends the systematic approach to curve sketching from Chapter 3.
• Curve-sketching software is helpful for visualizing the effects of changing a curve's equation.

Increasing, Decreasing, and Stationary Points

• A curve slopes upwards where the tangent creates a positive gradient, and y increases as x increases.
• A curve slopes downwards where the tangent creates a negative gradient, and y decreases as x increases.
• If f'(a) > 0, then f(x) is increasing at x = a.
• If f'(a) < 0, then f(x) is decreasing at x = a.
• If f'(a) = 0, then f(x) is stationary at x = a.

Stationary Points and Turning Points

• Stationary points, not on a constant function, are classified into four types: maximum turning point, minimum turning point, stationary point of inflection, or stationary point of inflection.
• A maximum turning point has the curve turn smoothly from increasing to decreasing with a maximum value at the point.
• A minimum turning point has the curve turn smoothly from decreasing to increasing with a minimum value at the point.
• A stationary point is a turning point if the derivative changes sign around it.
• Maximum turning point: The curve changes from increasing to decreasing.
• Minimum turning point: The curve changes from decreasing to increasing.
• At a point of inflection, the curve flexes, changes concavity, and the tangent crosses the curve.
• At a stationary point of inflection, the tangent is horizontal.
• A local or relative maximum occurs where the curve reaches a peak in its immediate vicinity.

Analyzing Stationary Points

• The derivative f'(x) can only change signs at a zero or discontinuity of f'(x), including stationary points of f(x) or where f(x) is non-differentiable.
• To analyze stationary points, find the zeroes and discontinuities of the derivative f'(x).
• Create a table of test values for f'(x) around its zeroes and discontinuities to observe gradient changes.
• The table reveals stationary point natures and function behavior across its domain, preparing for a sketch.

Less Familiar Curves

• The derivative y' of a function can only change sign at a zero or a discontinuity of the derivative.
• A discontinuity of y' can occur at a vertical asymptote.
• If a function y = f(x) has an asymptote at x = a, then the derivative y' is undefined there.

Analyzing Absolute Value Functions

• Absolute value functions are defined at some value x = a, but the derivative y' is not defined there.
• Sketch the graph of the function without calculus, and then understand the differentiation and table of slopes.

Second and Higher Derivatives

• The derivative of the derivative of a function is called the second derivative of the function
• Various notations are included such as f''(x), f^(2)(x), y'' and y^(2)

Concavity and Points of Inflection

• Concavity of a graph y = f(x) at x = a can be determined by the second derivative at x = a.
• If f''(a) is negative, the curve is concave down at x = a.
• If f''(a) is positive, the curve is concave up at x = a.
• A point of inflection is where the tangent crosses the curve and concavity changes sign around the point.

Using the Second Derivative

• Locate zeroes and discontinuities in the second derivative f''(x).
• Use a test values table for f''(x) around these points to determine concavity changes.
• The table indicates inflection points and the overall concavity of the graph.
• Finding the gradient of the tangent at each point of inflection is useful before sketching.
• These tangents are called inflectional tangents.
• f''(x) = 0 is not a sufficient condition for a point of inflection, because the sign of f''(x) must also change around the point.

Using the Second Derivative to Test for Maxima and Minima

• If a curve is concave up at a stationary point, the point is a minimum turning point.
• If a curve is concave down at a stationary point, the point is a maximum turning point.
• If f''(a) > 0, the curve is concave up at x = a, and there is a minimum turning point there.
• If f''(a) < 0, the curve is concave down at x = a, and there is a maximum turning point there.
• If f''(a) = 0, more work is needed by going back to the table of values of f'(x).

Summary of Curve Sketching Methods

• Domain: Find the domain of f(x).
• Symmetry: Find whether the function is even or odd, or neither.
• Intercepts: Find the y-intercept and all x-intercepts (zeroes).
• Sign: Use a table of test values of f(x), that is, a table of signs, to find where the function is positive, and where it is negative.
• Vertical Asymptotes: Examine any discontinuities to see whether there are vertical asymptotes there.
• Horizontal Asymptotes: Examine the behavior of f(x) as x approaches ∞ and -∞, noting any horizontal asymptotes.

Analyzing the Derivatives

• First Derivative:
  • Find the zeroes and discontinuities of f'(x).
  • Use a table of test values of f'(x), that is, a table of slopes, to determine the nature of the stationary points and the slope of the function throughout its domain.
• Second Derivative:
  • Find the zeroes and discontinuities of f"(x).
  • Use a table of test values of f"(x), that is, a table of concavities, to find any points of inflection and the concavity of the function throughout its domain.
• Any Other Features: A routine warning of incompleteness.

Global Maximum and Minimum

• Global (absolute) maximum: f(x) ≤ f(a) for all x in the domain.
• Global (absolute) minimum: f(x) ≥ f(a) for all x in the domain.
• Examine and compare: turning points, boundaries of the domain (or the behavior for large x), and discontinuities of f'(x).

Applications of Maximization and Minimization Problems

• Usually a diagram should be drawn. Then:
  • Introduce the two variables from which the function is to be formed.
  • Form an equation in the two variables, noting any restrictions.
  • Find the global maximum or minimum.
  • Write a careful conclusion.
• A claim that a stationary point is a maximum or minimum must always be justified by a proper analysis of the nature of the stationary point.

Primitive Functions

• This section reverses the process of differentiation.
• Many different functions may all have the same derivative.
• Any two functions with the same derivative differ only by a constant.
• If a function f(x) has derivative zero in an interval a < x < b, then f(x) is a constant function in a < x < b.
• If f'(x) = g'(x) for all x in an interval a < x < b, then f(x) and g(x) differ by a constant in a < x < b.

Finding a Function

• A function F(x) is called a primitive or an anti-derivative of f(x) if the derivative of F(x) is f(x), F'(x) = f(x).
• If F(x) is any primitive of f(x) then the general primitive of f(x) is F(x) + C, where C is a constant.
• To find a function, given its derivative and an initial condition:
  • First find the primitive, taking care to include the constant of integration.
  • Then substitute the known value of the function to work out the constant.

Finding Primitives

• dy/dx = x^n → y = (x^(n+1))/(n+1) + C where n cannot be -1.

Rules of Primitives

• If dy/dx = (ax+b)^n, where n is not -1, then y = (ax+b)^(n+1) / a(n+1) + C