# MCV4U – Grade 12 Calculus & Vectors – Curve Sketching

Grade 12 – Calculus and Vectors

Curve Sketching Test

Interval of Increase and Decrease

• A function is increasing if the slope of the tangent is positive over the entire interval
• A function is decreasing if the slope of the tangent is negative over the entire interval
• Intervals over which a function is increasing or decreasing can be determined by finding the derivative f’(x), and solving inequalities f’(x) <= 0 or f’(x) >= 0.

Maxima and Minima

• If f’(x) changes from positive to zero to a negative, then whenever it’s 0 will be a local maximum point
• If f’(x) changes from negative to zero to a positive, then whenever it’s 0 will be a local minimum point
• The absolute maximum and minimum values are found at local extrema or at the endpoints of the interval
• A critical number is a number in the domain of the function where f’(a) = 0 or f’(a) exists

Concavity and the second derivatives test

• The second derivative is the derivative of the first derivative. It is the rate of change of the slope of the tangent
• A function is concave up on an interval if the second derivative is positive on that interval
• A function is concave down on an interval if the second derivative is negative
• A function has a point of inflection at the point where the second derivative changes sign, where f’’(x) = 0
• Critical points can be classified by using second derivative or by examining the graph of f’’(x)
• If f’(a) = 0 and f’’(a) > 0, there is a local minimum at (a, f(a))
• If f’(a) = 0 and f’’(a) < 0, there is a local maximum at (a, f(a))
• If f’(a) = 0 and f’’(a) = 0 and f’’(x) changes sign at x = a, then there is a POI at (a, f(a))

Simple rational functions

• Vertical Asymptotes are the values of x that make the denominator equal to zero
• Consider VAs when finding intervals of concavity of increase or decrease

Curve Sketching

• When sketching a curve, remember to include domain, asymptotes, intercepts, critical values from the first derivative, second derivative critical values, local and absolute extremas, points of inflection, intervals of increase and decrease, and concavity.

Optimization problems

• Identify what the problem is asking
• Define variables and create a diagram if it helps
• Identify the quantity to be optimized and write an equation
• Define independent variable
• Define a function
• Identify and restrictions on the function
• Differentiate the function and classify critical points
• Check if the solution is whatever the question asked and if its in context with the question