Ch. 5 Applications of Derivatives

How do you find points of inflection?

1) Find critical points for the double derivative: Set the double derivative = 0 or find where it DNE. 2) Make a y'' number line and find out which intervals y'' is positive (concave up) and negative (concave down). 3) Once you find the x-values for POI, plug them back into the original function to find the y-values.

Show that the function f(x)=x² satisfies the MVT on the interval [0,2].

1) Find instantaneous ROC by taking a derivative f'(x)=2x, or f'(c)=2c 2) Find avg ROC between (0, f(0)) and (2, f(2)) f(2)-f(0)/2-0 = 4/2 = 2 3) Set instantaneous ROC = avg ROC and solve for c 2c=2 c=1 which is between 0 and 2, yay!

How do you find a and b, where y is increasing and decreasing?

1) Take the derivative of the function. 2) Set derivative = 0 and find where it DNE. These are your critical points. 3) Put critical points on a derivative number line. Find the intervals where y' is + (increasing) and - (decreasing). 4) Inc/Dec intervals MUST BE CLOSED, such as (-∞, 3]U[5,7].* *Only exception is when the x-value DNE in the original function.

What are the two requirements for f to have both an absolute max and min, according to the EVT?

1) continuous 2) on a closed interval

5.1 Extreme Values

5.1 Extreme Values

5.2 Mean Value Theorem

5.2 Mean Value Theorem

5.3 1st and 2nd Derivative Test

5.3 1st and 2nd Derivative Test

absolute (global) extreme values

Absolute max if f(c) ≥ f(x) for all x Absolute min if f(c) ≤ f(x) for all x

Concavity Test: The graph of a twice differentiable function f(x) is concave down when?

Any interval where y''<0

Concavity Test: The graph of a twice differentiable function f(x) is concave up when?

Any interval where y''>0

How do you do k, sketch the graph?

Combine all of the different information collected in A-J into one graph. Keep in mind x-ints, y-ints, increasing and decreasing, concavity, and end behavior.

How do you find e and f, where y is concave up and down?

Concavity means take DOUBLE DERIVATIVE y''. Concave up = like a cup Concave down = like a frown 1) Set y'' = 0 and find where it DNE- these are your critical points. 2) Create y'' number line with critical points, and plug in numbers to see when y'' is + or - on the intervals. 3) If y''>0: concave up. If y''<0: concave down. 4) Concavity uses OPEN INTERVALS, unlike increasing and decreasing.

f(x) = 3x². What is the antiderivative?

F(x)=x³ + C + C is where the infinitely many antiderivatives comes in. We do not know what C is.

6) If f''(2)<0, then x=c is a point of inflection for the function f and cannot be the x-coordinate of a maximum or minimum point on the graph of f.

False; A point is a POI only if the sign of f'' changes at that point.

8) The absolute minimum value of a continuous function on a closed interval can occur at only one point.

False; An absolute minimum value is a y-value, which multiple x-values can have. An example is y=sin x with a minimum of y=-1.

1) A critical point of a function f(x) is the x-coordinate of a relative maximum or minimum value of the function.

False; Critical points can also be stationary points, where there is no local min or max, such as x=0 in f(x)=x³.

10) To locate the absolute extrema of a continuous function on a closed interval, you need only compare the y-values of all critical points.

False; Extrema occurs at critical points and endpoints. The closed interval may not include critical points, but it does include endpoints.

3) If f''(x) is always positive, then the function f must have a relative minimum value.

False; For f(x)=e^x the function is always concave up but it does not have a relative min.

11) If f'(c)=0 and f'(x) decreases through x=c, then x=c locates a local minimum value for the function.

False; These conditions mean that x=c is a local maximum value, not minimum. Change min to max

4) If a function f has a local minimum value at x=c, then f'(c)=0.

False; f'(c) can also be undefined at a minimum value, like x=0 in y=IxI.

How do you find g, the point of inflection?

Find at what x-value on the y'' number line the concavity changes from up to down, or down to up. y'' must change signs. This indicates a switch in concavity = POI. Plug the POI x-value into the original equation to get an ORDERED PAIR as the point of inflection.

How do you find c and d, the relative max and min points?

For max, use the y' number line to see when y goes from increasing to decreasing (y' from + to -). Plug that x-value into the original equation. For min, use the y' number line to see when y goes from decreasing to increasing (y' from - to +). Plug that x-value into the original equation. Both points must be ordered pairs.

antiderivative

If F(x) is a function and f(x) is its derivative, then F(x) is the antiderivative of f(x).

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a,b], then f has both an ABSOLUTE maximum and an absolute minimum value somewhere on the interval.

Rolle's Theorem

If f(x) is a polynomial and f(a)=0 and f(b)=0 (x-ints), then there is at least one point c between a and b such that f'(c)=0. Meaning: If f(x) is a polynomial with two x-intercepts, a and b, then there is at least one point c that is an absolute max or min between a and b with a derivative of 0.

Mean Value Theorem (MVT)

If f(x) is continuous on a closed interval [a,b] and f(x) is differentiable at every interior point (a,b) then there is at least one point c in interior points (a,b) at which f'(c) = f(b)-f(a)/b-a. Meaning: If a function satisfies 3 conditions (a) continuous and b) differentiable on c) a closed interval [a,b]), then there is at least one point c in the interval at which the instantaneous ROC of c = avg ROC between a and b. AKA: Slope of secant line = slope of tangent line.

Corollary 2: If f'(x)=0 at each point of an interval I, then f(x)= some constant C for all x in I. What does this mean?

If slope = 0 for every point in an interval, then y=C (horizontal line).

Corollary 3: If the derivative of both f(x) and g(x) are the same, then f(x) and g(x) are the same function except for a vertical shift of c. What does this mean?

If the derivative of two nonidentical functions is the same, then f(x)=g(x) + C.

First Derivative Test for Local Extrema

Let c be a critical point (point where f'(c)=0 or DNE). Local max: Occurs at critical point c if f' changes sign from + to - (inc to dec) at c. Local min: Occurs at critical point c if f' changes sign from - to + (dec to inc) at c.

local (relative) values

Local max if f(c) ≥ f(x) for all x in some open interval containing c. Local min if f(c) ≤ f(x) for all x in some open interval containing c. Essentially, local extrema just have to be mins or maxes in comparison to the points around them.

2nd Derivative Test for Local Extrema

Local max: If f'(c)=0 and f''(c)<0 then f has a local MAX at x=c. Concave down = max. Local min: If f'(c)=0 and f''(c)>0 then f has a local MIN at x=c. Concave up = min.

How do you find j, the limits as x approaches -∞ and +∞?

Look at end behavior of the graph.

If you are finding the extrema of a function within open intervals, can endpoints be extrema?

No because they DNE

When asked to find extreme values of a function that has √ in the denominator, what is the domain?

Since there is a √, then the range must be ≥0. however, because √ is in the denominator and denominators cannot equal 0, √ must be >0. Set whatever is inside the √ as >0 and solve for the domain.

5) If f'(2)=0 and f''(2)<0, then x=2 locates a relative maximum value of f.

True

7) If a function f is defined on a closed interval and f'(x)>0 for all x in the interval, then the absolute maximum value of the function will occur at the right endpoint of the interval.

True

9) If x=2 is the only critical point of a function f and f''(2)>0, then f(2) is the minimum value of the function.

True

True or False: Classifying Critical Points (#1-12)

True or False: Classifying Critical Points (#1-12)

12) Absolute extrema of a continuous function on a closed interval can occur only at endpoints or critical points.

True!

2) A continuous function on a closed interval can have only one maximum value.

True; There can be only one maximum value, but it can be shared by more than one x-value.

True or false: There can be only one min/max value.

True; but the value can occur at multiple places

Where do functions fail to have a derivative?

Whenever derivative on the left ≠ derivative on the right. 1) Corner such as f(x) = IxI 2) Cusp such as f(x)=x^2/3 3) Vertical tangent such as f(x)=^3√x aka x^1/3 4) Discontinuity

How do you find h and i, the x and y intercepts?

X-intercepts: Set function = 0 to find what x is when y=0. Y-intercepts: Plug x=0 in and find y.

critical point

a point in the interior of the domain of a function f at which either f'=0 or does not exist

Write down the parts of an A-K Problem.

a) Over what intervals is y increasing? b) Decreasing? c) Find the relative max point. d) Find the relative min point. e) Where is y concave up? f) Where is y concave down? g) Find the point of inflection. h) Find the x-intercepts. i) Find the y-intercept. j) Find limit as x approaches ∞ and -∞. k) Sketch.

What are some qualities of a polynomial?

continuous with smooth curves

Extreme values (mins and maxes) occur only at what two kinds of points?

critical points and closed endpoints

If f'<0 then f

decreases

If slope of f is negative, then f

decreases

If c is a critical point but f does not change sign at c (f' has is inc or dec on both sides of c), then f...

has NO extreme value at c.

If f'>0 then f

increases

If slope of f is positive, then f

increases

How many antiderivatives does a function have?

infinitely many

point of inflection

point where the concavity changes