## Friday, July 3, 2020

### Is the Derivative of a Function the Slope

We must be cautious in calling the derivative of a function the slope. Ã‚  Although the two concepts are clearly related, there are nuances to each that separate them. Derivative and Slope:Ã‚  Whats the difference? Let us start with the definition of each. A derivative of a function is a representation of the rate of change of one variable in relation to another at a given point on a function. Ã‚   The slope describes the steepness of a line as a relationship between the change in y-values for a change in the x-values. Clearly, very similar ideas. Ã‚  But letÃ¢â‚¬â„¢s look at the important differences. Ã‚  A functionÃ¢â‚¬â„¢s derivative is a function in and of itself. Ã‚  It may be a constant (this will happen if our function is linear) but it may very well change between values of x. Let f(x) = x2. Ã‚  Our derivative fÃ¢â‚¬â„¢(x) = 2x. Ã‚  If we take a look at the graph of x2, we can see that for each step we take along the curve, the value of y changes more and more. Ã‚  Between x = 0 and x = 1, y only increases by 1. Ã‚  But between x = 1 and x = 2, y increases by 3. Ã‚  If we keep going with this trend, between x = 2 and x = 3, y changes by 5. Ã‚  We donÃ¢â‚¬â„¢t have a constant change between equally spaced values of x, but rather y changes by twice as much each step. A slope has the same idea, but can only be used for a line. Ã‚  The slope of a line tells us how much that lineÃ¢â‚¬â„¢s y value changes for any given change in x, but we do not use this term for curves or non-linear functions as by definition, our slope is constant: A line always has the same slope. Ã‚  Every step we take along the x-axis, the change in our value of y remains constant. Ã‚  A positive slope indicates that y increases as x increases. Ã‚  Ã‚  A negative slope implies that y decreases as x increases. Ã‚  And a 0 slope implies that y is constant. Ã‚  We cannot have the slope of a vertical line (as x would never change). A function does not have a general slope, but rather the slope of a tangent line at any point. Ã‚  In our above example, since the derivative (2x) is not constant, this tangent line increases the slope as we walk along the x-axis. Ã‚  We cannot have a slope of y = x2 at x = 2, but what we can have is the slope of the line tangent to this point, which has a slope of 4. We can also take multiple derivatives, each gives us a new piece of information about our curve. Ã‚  If the derivative of a function tells us how one variable changes with respect to another, the derivative of the derivative (named the second derivative or double derivative) tells us how about the change in the change of one variable with respect to another. Ã‚  If we take the example above y = x2, the derivative yÃ¢â‚¬â„¢ = 2x shows us that the slope of a tangent line is constantly increasing. Ã‚  The second derivative yÃ¢â‚¬â„¢Ã¢â‚¬â„¢ = 2 tells us that the change in this change is constant. This is easier to see in a physical representation. Ã‚  Let us give the position of a function as x(t) = 3t2-2t+1. Ã‚  We can see that the position is not linear. Ã‚  The derivative of this function xÃ¢â‚¬â„¢(t) = 6t -2 gives us our velocity at any give time. Ã‚  Our velocity we can see is also itself changing with time. Ã‚  If we take our second derivative xÃ¢â‚¬â„¢Ã¢â‚¬â„¢(t) = 6 shows us how our velocity is changing with time. Ã‚  This is named our acceleration, which in our above example is constant. It is important to remember how to use the derivative to find the slope of a tangent line, but remember that the derivative itself is not a slope in and of itself. Ã‚  The derivative is a powerful idea that use used in many different ways.

## Tuesday, May 26, 2020

### The Investing in the Shares of Home Depot due to Its Better Performance Compared to Lowes Inc Free Essay Example, 1000 words

From the data analysis conducted, some conclusions can be deduced, including the market growth of the Home Depot Company, the growth prospects, and the investment options available for shareholders. The sales data for the two companies indicate that Home Depot has a better market share that Lowe s Inc. This conclusion is arrived at after considering the fact that the sales for Home Depot are better compared to the sales for Lowe s; the sales for Home Depot are more than the sales of Lowe s for all years of operations. The net income as a percentage of sales also indicates that the company is better for Home Depot, which indicates that the company utilizes resources better than Lowe s Inc. Though the debt figures for Home Depot are higher than the debt figures for Lowe s, the debt-to-equity ratio indicates that Home Depot utilizes the ratio better that Lowe s Inc. The board of directors for Home Depot is made up of 10 executive directors, individuals with exceptional records of accomplishment in their fields of profession. The first opinion when the board of directors is analyzed is that the company is committed to excellence, because the members of the executive board all seem to be competent members. We will write a custom essay sample on The Investing in the Shares of Home Depot due to Its Better Performance Compared to Lowes Inc or any topic specifically for you Only \$17.96 \$11.86/page