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Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test

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## Section 3.3 – Increasing and Decreasing Functions and the First Derivative Test

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**Section 3.3 – Increasing and Decreasing Functions and the**First Derivative Test**How Derivatives Affect the Shape of the Graph**Many of the applications of calculus depend on our ability to deduce facts about a function f from in information concerning its derivatives. Since the derivative of f represents the slope of tangents lines, it tells us the direction in which the curve proceeds at each point. Thus, it should seem reasonable that the derivative of a function can reveal characteristics of the graph of the function.**Increasing and Decreasing Functions**The function f is strictly increasing on an interval I if f (x1) < f (x2) whenever x1 < x2. The function f is strictly Decreasing on an interval I if f (x1) > f (x2) whenever x1 < x2. D f(x) Decreasing Increasing B A C x**How the Derivative is connected to Increasing/Decreasing**Functions When the function is increasing, what is the sign (+ or –) of the slopes of the tangent lines? When the function is decreasing, what is the sign (+ or –) of the slopes of the tangent lines? POSITIVE Slope NEGATIVE Slope D f(x) B A C x**Test for Increasing and Decreasing Functions**Let f be differentiable on the open interval (a,b) If f '(x) > 0 on (a,b), then f is strictly increasing on (a,b). If f '(x) < 0 on (a,b), then f is strictly decreasing on (a,b). If f '(x) = 0 on (a,b), then f is constant on (a,b).**Procedure for Finding Intervals on which a Function is**increasing or Decreasing If f is a continuous function on an open interval (a,b). To find the open intervals on which f is increasing or decreasing: • Find the critical numbers of f in (a,b) AND all values (a,b) of x in that make the derivative undefined. • Make a sign chart: The critical numbers and x-values that make the derivative undefined divide the x-axis into intervals. Test the sign (+ or –) of the derivative inside each of these intervals. • If f '(x) > 0 in an interval, then fis increasing in that same interval. If f '(x) < 0 in an interval, then fis decreasing in that same interval. • State your conclusion(s) with a “because” statement using the sign chart. A sign chart does NOT stand on its own.**Example 1**Use the graph of f '(x) below to determine when fis increasing and decreasing. f is increasing when the derivative is positive. f ' (x) f is increasing when the derivative is positive. x f is decreasing when the derivative is negative. Increasing: (-1,3) (-∞,-1) U (3,∞) Decreasing:**White Board Challenge**The graph of f is shown below. Sketch a graph of the derivative of f.**White Board Challenge**Find the critical numbers of:**Example 2**Find where the function is increasing and where it is decreasing. Domain of f: All Reals Find the critical numbers/where the derivative is undefined Find the derivative. Find where the derivative is 0 or undefined Find the sign of the derivative on each interval. -1 2 0 Answer the question**Example 2: Answer**The function is increasing on (-1,0)U(2,∞) because the first derivative is positive on this interval. The function is decreasing on (-∞,-1)U(0,2) because the first derivative is negative on this interval.**Example 3**Use the graph of f (x) below to determine when fis increasing and decreasing. f is decreasing when the function’s outputs are getting smaller as the input increases. Notice how the function changes from increasing to decreasing at x=-3. But since -3 is not in the domain of the function, it is not a critical point. Thus, critical points are not the only points to include in sign charts. f (x) f is increasing when the function’s outputs are getting larger as the input increases. x Increasing: (-3,∞) (-∞,-3) Decreasing:**Example 4**Find where the function is increasing and where it is decreasing. Domain of f : All Reals except -1 Find the critical numbers/where the derivative is undefined Find the derivative. Find where the derivative is 0 or undefined The function does not have any critical points: the derivative is never equal to 0 and the derivative is only undefined at a point not in the functions domain (x=-1). Even though -1 is not a critical point, it can still be a point where a function changes from increasing to decreasing. ALWAYS include every x value that makes the derivative undefined on a sign chart (even if its not a critical point). Find the sign of the derivative on each interval. -1 Answer the question**Example 4: Answer**The function is decreasing on (-∞,-1)U(-1,∞) because the first derivative is negative on this interval. Make sure not to include -1 in the interval because it is not in the domain of the function.**White Board Challenge**Find the maximum and minimum values attained by the given function on the indicated closed interval:**Critical Values and Relative Extrema**Remember that if a function has a relative minimum or a maximum at c, then c must be a critical number of the function. Unfortunately not every critical number results in a relative extrema. A new calculus method is needed to determine whether relative extrema exist at a critical point and if it is a maximum or minimum.**How the Derivative is connected to Relative Minimum and**Maximum When a critical point is a relative maximum, what are the characteristics of the function? When a critical point is a relative minimum, what are the characteristics of the function? The function changes from increasing to decreasing The function changes from decreasing to increasing D f(x) B A C x**The First Derivative Test**Suppose that c is a critical number of a continuous function f(x). (a) If f '(x) changes from positive to negative at c, then f(x) has a relative maximum at c. f(x) Relative Maximum f '(x) > 0 f '(x) < 0 c x**The First Derivative Test**Suppose that c is a critical number of a continuous function f(x). (b) If f '(x) changes from negative to positive at c, then f(x) has a relative minimum at c. f(x) f '(x) < 0 f '(x) > 0 Relative Minimum c x**The First Derivative Test**Suppose that c is a critical number of a continuous function f(x). (c) If f '(x) does not change sign at c (that is f '(x) is positive on both sides of c or negative on both sides), then f(x) has no relative maximum or minimum at c. f(x) f(x) f '(x) > 0 f '(x) < 0 No Relative Maximum or Minimum f '(x) < 0 f '(x) > 0 c x x c**Example 1**Use the graph of f '(x) below to determine where fhas a relative minimum or maximum. Find the Critical Numbers f ' (x) Make a sign chart and Find the sign of the derivative on each interval. x Apply the First Derivative Test. -1 3 Relative Maximum: @ -1 @ 3 Relative Minimum:**Example 2**Find where the function on 0≤x≤2π has relative extrema. Domain of f: 0≤x≤2π Find the critical numbers Find the derivative. Find where the derivative is 0 or undefined Find the sign of the derivative on each interval. Find the value of the function: 0 2π/3 2π 4π/3 Answer the question**Example 2: Answer**The function has a relative maximum of 3.826 at x = 2π/3 because the first derivative changes from positive to negative values at this point. The function has a relative minimum 2.457 at x =4π/3because the first derivative changes from negative to positive values at this point.**Example 3**Domain of f: Find the relative extrema values of . All Reals Find the critical numbers Find the derivative. Find where the derivative is 0 or undefined The derivative is undefined at x=-3,0 Find the sign of the derivative on each interval. -3 0 -1 NOTE: 0 is not a relative extrema since the derivative does not change sign. Answer the question**Example 3: Answer**First find the value of the function: Now answer the question: The function has a relative maximum of 0 at x = -3 because the first derivative changes from positive to negative values at this point. The function has a relative minimum of -1.587 at x = -1 because the first derivative changes from negative to positive values at this point.**White Board Challenge**BONUS: How many critical numbers are there? Find the intervals on which the function below is increasing or decreasing.