Find the derivative of (cos(x))^3
To calculate this derivative, the chain rule must be used. For the chain rule, you always start with the outermost part of the expression. In this case, the cos(x) is all raised to the third power. Think of the cos(x) as one unit. We will define is as u. So now, we have u^3. The derivative of this is a simple power rule which results in 3u^2. Now, we have to multiple 3u^2 by the derivative of u to complete the chain rule. The derivative of u, which is really cos(x) is sin(x). This means that our final is 3u^2*sin(x). But we still have one more step. We must plug in cos(x) for u because that is what we defined it as. Our final answer becomes 3*(cos(x))^2*sin(x)
What are some important design considerations for the design of a new heart valve?
A heart valve must be biocompatible. This means that the material that the heart valve is constructed of must not be toxic to the body or produce an adverse immune response. For example, various metal can release toxic ions. Biological materials from other species could have proteins on the surface that our body recognizes as foreign and attacks. Another consideration for a heart valve is making sure the material does not cause blood clotting. The valve should also not alter flow patterns within the heart. Altering flow patterns can change the stresses on the wall of the heart. By increasing stresses, the heart walls could become enlarged and thicker. There are many more considerations including fatigue, sewing ring concentrations, minimally invasive approaches, and the noise the valve makes.
Explain the difference between engineering stress and true stress.
When a force in applied to an object, deformations result. True stress is equal to the force applied to a material divided by the instantaneous area over which the force is applied. The key word here is "instantaneous." To visualize, as a material is pulled the material will deform and get longer, causing the cross sectional area to get smaller as the force increases more and more. This means that to calculate true stress you need to know the different area at each and every time point, which is essentially infinite areas. This makes true stress very difficult to calculate. That's where engineering stress comes in. Engineering stress is the force applied to a material divided by the initial area. This means that to calculate engineering stress, you only need to take one area measurement at the beginning before force is applied. Necking of a material is when the middle of the material becomes smaller as it is being pulled. Before necking, engineering and true stress are the same value because the cross sectional area is not changing from the initial area. After necking occurs in a material, engineering stress and true stress have different values. This concept can be further visualized with a stress vs strain curve.