CURVE SKETCHING The first application of derivatives we will study is using derivatives to determine the shape of the graph of a function. We will use the derivative to determine when a function is increasing or decreasing. We will use the derivative to determine where a function changes direction or has extremes. We will use derivatives to determine how the function curves over intervals of the domain. There is no need to plot points in order to understand the general shape of the graph.

Increasing / Decreasing We know from studying linear functions that a function is increasing when its slope (or rate of change) is positive. The derivative is the instantaneous rate of change. To determine if a function is increasing or decreasing, first find its derivative f’(x) > 0function is increasing f’(x) < 0 function is decreasing

EXTREMES In Pre-Calc, we defined the extremes as the places where the function changes direction This is still true for functions over an open interval (a, b) These were the maximum and minimum values in a local region For a function over a closed interval [a, b], we need to look at the endpoints and determine if those represent a maximum or a minimum in those regions.

CRITICAL POINTS Extremes can only occur at endpoints and at critical points. They do not have to occur at these points but it is only at these points that they can occur. A critical point is a location where the derivative of the function is undefined or equal to zero. f’(x) = 0 f’(x) = undefined A max or minimum will have a derivative of zero A cusp or corner will have an undefined derivative

Finding Critical Points

1 ST Derivative Test f’(x) Behavior of f(x) Let f be continuous on (a, b) Locate the critical points of f on (a, b) Use these points to determine test intervals Determine the sign of f’ in each test interval f’(x)>0: increasingf’(x)<0: decreasing Address the behavior of the function at the critical points

1 st Derivative Test This method proves the behavior of the function over all the intervals of the domain It also proves the extremes: –Maximums change from increasing to decreasing –Minimums change from decreasing to increasing –If the function does not change direction at these points, there is no extreme at this point. This method finds the location of the extremes which is how we denote their existence. The AP test will often ask for the “value” of the extremes – this is the value of the function or y at that value of x. Plug in for x, solve for y

LOCAL vs ABSOLUTE EXTREMES The 1 st derivative test lets us find the “local” extremes: the maximum or minimum value of the function in that region of the domain If the function changes direction here, it reaches an extreme. Absolute extremes involve the actual maximum or minimum value of the function across the domain. To determine these, you will have to know the value of the extreme if there is more than one or to know that the function is not bounded above (maximum) or below (minimum)

CLOSED INTERVALS Earlier we said that an extreme could occur at the endpoints. Endpoints occur when you have a closed interval [a, b] In order to determine if there is a local max or min at the endpoints, we look again to the derivative. Left Endpoint: f’(x)>0 the function is increasing as it moves to the right meaning the left endpoint is a minimum. It is a maximum if the derivative was negative Right Endpoint: f’(x)>0, the function was increasing to this point making it a maximum. A minimum occurs at the right endpoint if the derivative was negative. These can be indicated on the 1 st derivative chart.

ROLLE’S THEOREM If the function is continuous at every point of a closed interval [a, b] and it is differentiable at every point (a, b) and f(a) = f(b), then there is at least one point, x = c, where the instantaneous rate of change (dervative) is equal to zero.

MEAN VALUE THEOREM If the function is continuous at every point of a closed interval [a, b] and it is differentiable at every point (a, b) then there is at least one point where the instantaneous rate of change (slope) is equal to the average rate of change (slope).

FIRST DERIVATIVE Tells the shape of the graph of the function. At the a critical point: f’ changes from positive to negative: local max f’ changes from negative to positive: local min f’ does not change sign: no extreme At left endpoint: f’ 0:local min At right endpoint: f’ 0:local max

USING 1 ST DERIVATIVE Build a table to illustrate the sign of the first derivative over different intervals. Find the critical points f(x) = x 3 – 12x – 5 f’(x) = 3x 2 – 12f ‘(x) = 0x = -2, 2 Interval - ∞ < x < < x < 2 2 < x < ∞ Sign of f’ Behavior of f(x)

PRODUCTS If your derivative is a product, look at the sign of each factor to determine the sign of the derivative f(x) = (x 2 - 3)e x f’(x) = f ‘(x) = 0x = Interval Sign of f’ Behavior of f(x)

GRAPH OF THE FUNCTION Look at the first example, f(x) = x 3 – 12x – 5 We have found the 1 st derivative and located the extremes at : x = -2, 2 We have also determined from our table which is a max and which is a min but does the graph of the function look like between those points? How do you know if it is a straight line or curved? If it is curved, what does it look like?

CURVE OF THE GRAPH We know how to find the extremes of the function – even the global extremes We do not know how it gets from one extreme to the next We know how to find when the function is increasing or decreasing. We do not know if the function is increasing at an increasing rate (getting steeper) or increasing at a decreasing rate (getting flatter) Derivative will help with this as well.

2 nd Derivative Test The second derivative is the rate of change of the first derivative. It tells how the slope is changing. If the function is increasing it has a positive slope. If the 2 nd derivative is also positive, the slope is becoming more and more positive or steeper. Increasing at an increasing rate. Conversely, if the 2 nd derivative is negative it make the slope less positive or flatter. Increasing at a decreasing rate.

2 nd Derivative Test The same thing will happen to a decreasing function – just in the opposite direction. A negative 2 nd derivative will make it get steeper in the negative direction. A positive 2 nd derivative will make it get flatter in the negative direction. If the 1 st and 2 nd derivatives have the same sign, the graph of the function will get steeper If the 1 st and 2 nd derivatives have opposite signs, the graph will get flatter

CONCAVITY Concavity describes a curve Concave means that the curve could hold water like a bowl Concave down means that it will not hold water. A function that is increasing at an increasing rate gets steeper – concave up The second derivative tells whether a function is concave up or down. f’’(x) > 0positiveconcave up f’’(x) < 0negativeconcave down

2 nd Derivative Test for Extremes The 2 nd derivative can also prove if an extreme is a maximum or minimum. A maximum must be concave down so if the 2 nd derivative at a critical point will be negative A minimum must have a positive 2 nd derivative because it is concave up.

POINTS OF INFLECTION A point of inflection is where the function is differentiable and the concavity changes. A point of inflection will occur where the 2 nd derivative will be either = 0 or undefined. At a point of inflection the concavity changes from up to down or down to up. If the concavity does not change, there will NOT be an inflection point at this location At a point of inflection, the graph of f’(x) will have a maximum or minimum.

CONCAVITY AND LINEARIZATION Remember, we used the tangent line of a function to approximate its value over small intervals of the domain. The tangent line of a function that is concave up will always lie below the actual function. –This means that the linear approximation will underestimate a function that is concave up. Of course the opposite happens to a function that is concave down – its tangent line will lie above the actual function and thus overestimate the value of the function

f(x) = 2x 3 – 14x x – 5 x ≥ 0 1st Derivative: f’(x) = 0 2 nd Derivative: f”(x) = 0 Interval Sign of f’’ Behavior of f(x) Sign of f’’ Behavior of f(x)

1 st Derivative tells when the function is increasing and decreasing. 1 st Derivative finds the locations of the extremes 2 nd Derivative tells when the function is concave up or down 2 nd Derivative finds the locations of the inflection points. 2 nd Derivative will tell if an extreme is a max or a min

MULTIPLICITY OF FACTORS Factoring is very useful in the 1 st and 2 nd derivative tests. It is easy to determine the sign during an interval by using the factors. Remember parabolas of perfect squares and how they sit on the x-axis. The function did not cross the x-axis when we had a an even number of factors. This will happen in these cases as well, if there is an even number of factors that determine a critical point – it will not change sign there.

CURVE SKETCHING Identify the intercepts –y-intercept: plug in x = 0 –x-intercept: solve for the zeros set factors = 0 determine when the numerator = 0 exponential function never equals zero unless vertically shifted down Identify any symmetry Identify discontinuities –removable – hole in the graph – discontinuity can be simplified away –infinite – vertical asymptote – discontinuity cannnot be simplified away

CURVE SKETCHING y = 2(x 2 – 9) / (x 2 – 4) y’ = 20x / (x 2 – 4) 2 Identify Asymptotes –Vertical asymptotes – infinite discontinuities log functions have vertical asymptotes –Horizontal asymptotes Rational functions end behavior Exponential functions Identify Extremes Identify Increasing / Decreasing Identify Concavity / Inflection Points

f(x) = 2x 5/3 – 5x 4/3