Link Concave

Link Concave

could you help me with these math problems(concave)?

the graph for these questions is on this link:

http://www.lakesideschool.org/upperschool/departments/math/ballard/xcalc/AP/97AP/97APsec2.pdf

3.
A) for what intervals of t is G(t) increasing and concave down
B) for what intervals of t is decreasing?

5.
for which values of t is G(t) > 1

7.
true or false?(explain)
The abolute value of G(t) is -1

3a) Concave means dished inward, so concave down means that the underside of the curve is dished inward. In this example, we aren't looking at a graph of the function itself but its derivative. If the function is concave downward and differentiable, then its second derivative is negative. This means the derivative of the derivative will have a negative slope. In the graph of f(x), the slope is negative at 0 < x < 2 and 2 < x < 3. You might be able to argue that the function is also concave downward at x = 2 despite the infinite discontinuity in the derivative of f(x) at x = 2, because the slope of f(x) is somehow negative even though its value is not defined, but it won't be easy.

3B) If G(x) is decreasing, then f(x), which is the derivative of G(x), must be negative. This happens over the interval 2 < x < 4.

5) Finding the exact value will require integrating the function that is a half-circle centered at (0, 0) and with a radius of 2. For -2 ≤ x ≤ 2, the function f(x) = 2√[1 - (x/2)²]. Making the trigonometric substitution x = 2 sin θ, we get x/2 = sin θ, dx = 2 cos θ dθ, and
G(x) = ∫2√[1 - (t/2)²] dt , -2 ≤ t ≤ 2, 0 ≤ x ≤ x0, where G(x0) = 1
= ∫4 cos² θ dθ, -π/2 ≤ θ ≤ π/2
=2 ∫1 + cos 2θ dθ, -π/2 ≤ θ ≤ π/2
= 2θ + sin 2θ (In this instance, the constant of integration is 0)
= 2θ + 2 sin θ cos θ
= 2 arcsin (x/2) + 2x√[1 - (x/2)²]

x0 is therefore the solution to the equation
1 = 2 arcsin (x0/2) + 2x0√[1 - (x/2)²]
I don't think this is solvable in closed form.
Nevertheless, it can be shown by constructing a rectangle with its lower left corner at (0, 0) and its upper corner at (1/2, 2), which has an area of 1, that G(1/2) < 1 By constructing another rectangle with lower left corner at (0, 0) and its upper corner at (1, √3), which is on the function, it can be shown that the area of this rectangle is √3 and the area under the curve is greater than that, so G(1) > √3. We therefore know that 1/2 < x0 < 1. If it is necessary to solve for this numerically, we can.

The area under the circular quadrant will be 1/4π*2² or π, so this is the value of G(2). The area under the negative part of the curve between x = 2 and x = 4 is the sum of two right triangles with base 1 and height of -1, for a total area under this curve. The function G(x) will reach a minimum value at x = 4, which will be the area of the quarter circle minus the area of these two right angles, so G(4) = π - 1, which is still greater than 1. For values of x greater than 4, G(x) will increase even more, so G(x) stays greater than 1 for all x > x0.

7) Absolute values can never be less than zero, so we don't even have to look at the expression to tell that the absolute value of G(t) is never -1.

Health Link: Concave Chest / Sunken Chest

Torn Rotator Cuff

 

In an adult population, rotator cuff tears are the most common cause of debilitating shoulder pain and disability, with approximately 300,000 rotator cuff surgeries performed annually in the United States.

What is the anatomy and function of the rotator cuff?

The rotator cuff consists of four muscles (namely the supraspinatus, infraspinatus, teres minor, and subscapularis) that act in concert to both stabilize and move the shoulder joint. Due to the function of these muscles, sports which involve a lot of shoulder rotation – for example, serving in tennis, pitching in baseball, swimming, kayaking – often put the rotator cuff muscles under a lot of stress. ?

The diagnosis and management of rotator cuff disease places a significant financial burden on the U.S. economy, amounting to an annual cost of 3 billion dollars.

These muscles arise from the shoulder blade and insert on the humeral head to create a continuous cuff around the shoulder joint, and provide a link from the trunk to the arm. The ball (humeral head) and its socket (glenoid) have relatively little inherent stability, and have often been compared to a golf ball resting on a golf tee. In this capacity, an intact rotator cuff is essential to provide stability to the joint by a compressing the humeral head into the concave glenoid. A large torn rotator cuff , particularly of the subscapularis, can render the joint at risk for instability and dislocation.

The deltoid and rotator cuff muscles work synergistically to maintain a balance of forces around the shoulder joint in every direction. The deltoid and infraspinatus/teres minor maintain a balance in the vertical plane, while the subscapularis and infraspinatus balance each other in the horizontal plane. With lifting of the arm, the deltoid generates an upward force that is resisted by the downward force produced by the rotator cuff muscles, preventing a loss of reduction of the humeral head on the glenoid. A torn rotator cuff can disrupt this balance of forces and compromise normal shoulder joint motion. In fact, a high riding humeral head that shifts superiorly off the glenoid with raising of the arm can be seen in the setting of a massive torn rotator cuff.

What is a torn rotator cuff?

A torn rotator cuff is a disruption in the integrity of the tendon at the insertion into the humeral head. Tendons connect the rotator cuff muscle belly to bone. Most commonly, tears involve the supraspinatus tendon but can involve any combination or all four of the rotator cuff tendons. The mechanism of injury can be highly variable. A torn rotator cuff can result from trauma such as a fall on the shoulder or after a shoulder dislocation. More commonly, however, athletes suffer a torn rotator cuff from repetitive wear and tear activities that strain and chronically fail the tendon. Such tears are particularly prevalent in overhead athletes and are often seen in tennis players, baseball pitches, javelin throwers, swimmers, and football quarterbacks. Sometimes, a narrow space for passage of tendon underneath the acromion can result in direct mechanical abrasion of the tendon. This has been termed outlet impingement and is commonly referred to as impingement syndrome. A prominent acromial spur and thickened bursal tissue in the subacromial space can abrade the tendon running underneath.

How does a torn rotator cuff occur in athletes?

Damage and ultimately tearing of the rotator cuff tendons has been attributed to either static or dynamic causes. Static changes refer to impingement and mechanical abrasion of the tendons from narrowing of the subacromial space, most commonly due to roughness or “spurring” on the underside of the acromion or thickening of the coracoacromial ligament. On the other hand, a torn rotator cuff can result from abnormal dynamic motion of the humeral head and cuff relative to scapula, leading to abnormal strain on the tendon and tearing on either the joint or bursal side. For example, muscle weakness can allow the humeral head to rise higher towards the acromion and is considered to be one of the most common dynamic causes of a torn rotator cuff in athletes.

A torn rotator cuff result when the muscles and tendons of the rotator cuff become frayed under the acromion bone of the shoulder. This occurs both with aging as well as in younger people who perform repetitive overhead activities. Baseball pitchers as well as occupations that require overhead work are two examples of people at risk of sustaining rotator cuff tears. Pedro Martinez and Randy Johnson are both examples of professional, hall-of-fame bound pitchers who developed a torn rotator cuff.

What is the natural history of rotator cuff tears in athletes?

Many patients with rotator cuff tears are asymptomatic. As many as 50% of people over the age of 60 years may have rotator cuff tears. Correspondingly, however, many patients with shoulder pain may not have a cuff tear. In addition, the presence of a rotator cuff tear in a patient with shoulder pain does not necessarily mean that the tear is the primary cause of pain. It is clear, however, that patients with asymptomatic tears are at a high risk for symptom progression over time.

Unfortunately, rotator cuff tears do not heal spontaneously. In addition, tear size progresses over time and can unfortunately lead to irreversible changes in the tendon and muscle. Retraction of the tendon, scar formation, and atrophy of the muscle with infiltration of fat are all predictable changes that occur with greater chronicity of tears. These changes not only produce a weak shoulder with abnormal mechanics, but also compromise the ability to perform a surgical repair of the tendon to bone. The condition can progress to the point where the relationship between the humeral head and glenoid is permanently altered, with significant upward migration of the head. Arthritis secondary to a massive rotator cuff tear can develop as the humeral head erodes the superior glenoid and undersurface of the acromion. For throwing athletes, fixing a rotator cuff tear is important for them to retain their velocity and control of the ball. Pedro Martinez was able to return to the major league level of baseball competition after a repair of his full-thickness tear.

Symptoms of Rotator cuff tear.
What does a torn rotator cuff feel like?

While rotator cuff tears may be asymptomatic, they will frequently manifest as shoulder pain, particularly at night and during activities of daily living. Patients may complain of varying degrees of shoulder weakness and variable losses of range of motion. Crepitus and swelling can occur as well. On physical exam, patients with longstanding tears may have visible atrophy of muscles around the scapula. Functional deficits often correlate with the location of the tear. Overhead activities are often the most difficult and painful.

Overhead athletes with rotator cuff tears my complain of stiffness and pain during warm-up exercises. Pain is often most prominent during the acceleration phase of throwing or serving. Pitchers will often complain of a loss of velocity or ability to “control their pitch” at the mound.

What are my treatment options?

Treatment options for a rotator cuff tear can be broadly categorized as nonsurgical or surgical interventions.

Nonsurgical options offer the advantage of avoiding the complications of surgery, and focus on pain relief and improving function by increasing the compensatory role of surrounding muscles. On the other hand, nonsurgical treatment will not result in healing of the torn tendon. Correspondingly, the risk of recurrent symptoms as well as tear progression with irreversible, chronic changes is substantial.

Surgical repair offers the potential benefit of expeditious pain relief and cessates tear progression and secondary chronic changes. Improvements in surgical techniques allow the vast majority of rotator cuff tears to be addressed arthroscopically through minimally invasive techniques. Nonetheless, small risk of infection and stiffness after surgery exist.

What are the nonsurgical modalities available for a torn rotator cuff?

Rotator cuff exercises for rehabilitation:

Exercise is the most important and useful intervention in the nonoperative management of a torn rotator cuff. Initially, athletes should rest and avoid any provocative maneuvers that elicit discomfort. When the pain has resolved, stretching can begin. The initial focus is on obtaining full and painless range-of-motion. When this has been gradually achieved, strengthening of the intact rotator cuff muscles and associated peri-scapular musculature can ensue. Strengthening of the rhomboids, levator scapulae, trapezius, and deltoid is of tantamount importance to provide a stable platform to maximize the efficiency and function of the remaining, intact cuff tissue. Some useful rotator cuff exercises to focus on these muscles include:

Seated rows
Latissimus pull downs

Corticosteroid Injection:

Local steroid injections in the subacromial space can function as potent anti-inflammatory agents in the subacromial bursa. They are very effective in relieving night pain and can also be used as an augment to rehabilitation exercises in patients that otherwise cannot comply secondary to discomfort. Steroids can have adverse effects on the quality of tendon tissue and healing, however, and for this reason should not be performed more than 3 times in the same shoulder and no more frequently than at least 3 months apart.

Nonsteroidal Anti-inflammatory Drugs (NSAIDS):

NSAIDs help to both control inflammation and relieve pain, and can be very useful as an adjunct to rehabilitation exercises in the management of a torn rotator cuff. These medications can have significant gastrointestinal and renal side effects, however, and should be carefully monitored by a medical physician.

What does surgery involve for a torn rotator cuff?

Several factors influence the decision to pursue surgical treatment, including tear size and pattern, patient expectations, medical comorbidities, and occupational demands. Surgery for rotator cuff tears may be performed as an open, mini-open, or entirely arthroscopic procedure.

What is involved in postoperative rehabilitation?

In the postoperative period, the arm must be protected. The forces related to daily activities with the shoulder exceed the strength of the repair and can disrupt it until some healing has occurred. A postoperative brace maintains the arm in approximately 15 degrees of abduction and prevents any overhead activity. Ice packs or custom devices that circulate cooled fluid are very useful to control pain and swelling in the immediate postoperative period.

As pain resolves, early passive range-of-motion is initiated within the first week of surgery. A physical therapist can be a very useful adjunct to this process in order to maintain a safe, supervised program. Gentle pendulum exercises in the sling, as well as passive motions in forward flexion and external rotation are continued for the first six weeks.

Strengthening Exercises. Strengthening exercises are typically delayed until 8 to 12 weeks when healing has progressed and full range-of-motion has been achieved. Start strengthening exercises only after you have your health professional's approval. Muscle strengthening with rubber tubing can be very effective and often safer than weight machines. Strengthening of the scapular stabilizers (deltoid, trapezius, rhomboids, levator scapulae, etc) is paramount to the strengthening of the rotator cuff to maintain a stable platform and favorable posture for cuff mechanics. Patients continue strengthening for up to a year or longer until satisfactory strength and function are achieved. The degree of strength achieved often relates with the severity and chronicity of the initial tear.

Rehab after rotator cuff surgery can vary widely, but there are some general principles that are true for most patient having surgery for treatment of a torn rotator cuff. Usually these exercises are started gradually as soon as you can do the exercise routine without pain.

Check out the following link below as it provides 8 weeks of specific rehab videos of Rotator Cuff Exercises

The videos are:

 Beginning Exercises. 2 - 4 weeks
 The Pendulum. 2 - 4 weeks
 Stretching - Internal Rotator. 4 - 6 weeks
 Stretching - External Rotator. 4 - 6 weeks
 Pulley Exercises. 6 - 8 weeks
 Bench & Reach. 8 weeks
 External Rotation. 8 weeks
 Lat Pull-Downs. 8 - 10 weeks

How long will it take for me to get back to my sport?

Just like all tears are not created equal, neither is the rehabilitation and recovery. Unfortunately, these timelines need to be individualized based on the severity of the tear and demands of your sport. Tennis, baseball, and other overhand sports can be very demanding, particularly at high-levels of competition. With these sports, a gradual return to activity is planned with your doctor. After healing has occurred, this is usually initiated through a graduated and supervised throwing program in which distance and velocity is slowly increased as tolerated over 2 to 3 months. Small or partial thickness tears will generally permit an accelerated recovery compared to large tears, but the ultimate plan to get you back on the field or court must be determined by your doctor and should reflect a balance of moving forward expeditiously without placing the repair at undue risk.

Can I prevent a torn rotator cuff? Rotator cuff Exercises.

The etiology of rotator cuff tears is multifactorial, and it is unclear with current evidence if tears can be completely prevented. Maintaining the health of the rotator cuff muscles and peri-scapular musculature, however, can certainly help to prevent injury and optimize the kinematics of the shoulder joint. These strategies are employed by elite pitchers and overhead athletes who place remarkable demands on their shoulder and rotator cuff daily. The internal rotators are inherently stronger than the external rotator cuff muscles, and maintain a balance of these forces is important. Rotator cuff exercises to consider include:

Seated rows
Latissimus pull downs
Resisted tubing exercises
Side-lying external rotator
Propped external rotator
Shoulder roll
Shoulder blade squeeze
Wall push-ups

For more helpful information on “sports injuries” and issues, please visit SportsMD Media Inc. SportsMD is the most trusted resource for sports health and fitness information for people engaged in sports everywhere. We have assembled the sports industry’s leading Doctors and health experts – each sharing valuable, practical advice to keep you playing injury-free.

References:

Gartsman GM, Khan M, Hammerman SM. Arthroscopic repair of full-thickness tears of the rotator cuff. J Bone Joint Surg Am 1998;80:832-840.

Gerber C, Schneeberger AG, Beck M, Schlegel U. Mechanical strength of repairs of the rotator cuff. J Bone Joint Surg Br 1994;76:371-380.

Altchek DW, Warren RF, Wickiewicz TL, Skyhar MJ, Ortiz G, Schwartz E. Arthroscopic acromioplasty: Technique and Results. J Bone Joint Surg Am 1990;72:1198-1207.

Neer CS II. Anterior acromioplasty for the chronic impingement syndrome in the shoulder: A preliminary report. J Bone Joint Surg Am 1972;54:41-50.

About the Author

Dr. Asheesh Bedi is an Assistant Professor of Sports Medicine and Shoulder Surgery at the University of Michigan and MedSport Program. He is a team physician for the University of Michigan Athletic Department and specializes in both arthroscopic and open surgery for athletic injuries of the shoulder, elbow, hip, and knee.

Dr. Bedi completed his undergraduate training at Northwestern University where he graduated Summa Cum Laude. He graduated from the University of Michigan Medical School with AOA recognition, and remained in Ann Arbor to pursue residency training in Orthopaedic Surgery at the University of Michigan. After completing his training, Dr. Bedi completed a two-year fellowship in sports medicine and shoulder surgery at the Hospital for Special Surgery and Weill Cornell Medical College in New York. He has also pursued additional dedicated training with Dr. Bryan Kelly in arthroscopic hip surgery for young athletes. While in New York, he was an assistant team physician for the New Jersey Nets professional basketball and New York Mets professional baseball organizations with Dr. Riley Williams, Struan Coleman, and David Altchek. He was also an orthopaedic consultant for the U.S. Open Tennis Tournament in 2007 and 2008 with Dr. David Dines and an assistant team physician for Iona College Athletic Programs.