calculus examples of short term and long term

Calculus examples of short term and long term drivers#
Calculus examples of short term and long term driver#

To determine whether the spectators are in danger in this scenario, find the x-coordinate of the point where the tangent line crosses the line y = 2.8.

Find the equation of the tangent line to the curve at this point.Find the ( x, y ) ( x, y ) coordinates of this point near the turn.

Calculus examples of short term and long term drivers#

Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1.(b) The front corner of the grandstand is located at ( −1.9, 2.8 ).

Calculus examples of short term and long term driver#

We want to determine whether this location puts the spectators in danger if a driver loses control of the car.įigure 3.21 (a) One section of the racetrack can be modeled by the function f ( x ) = x 3 + 3 x 2 + x. The plans call for the front corner of the grandstand to be located at the point ( −1.9, 2.8 ). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. One section of the track can be modeled by the function f ( x ) = x 3 + 3 x 2 + x f ( x ) = x 3 + 3 x 2 + x ( Figure 3.21). Suppose you are designing a new Formula One track. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack. Normally, this just results in a wider turn, which slows the driver down. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. = 3 x 2 Let h go to 0.įigure 3.20 The grandstand next to a straightaway of the Circuit de Barcelona-Catalunya race track, located where the spectators are not in danger. = lim h → 0 ( 3 x 2 + 3 x h + h 2 ) After cancelling the common factor of h, the only term not containing h is 3 x 2. = lim h → 0 h ( 3 x 2 + 3 x h + h 2 ) h Factor out the common factor of h. = lim h → 0 3 x 2 h + 3 x h 2 + h 3 h In this step the x 3 terms have been cancelled, leaving only terms containing h. All other terms contain powers of h that are two or greater.

$calculus examples of short term and long term$

d d x ( x 3 ) = lim h → 0 ( x + h ) 3 − x 3 h = lim h → 0 x 3 + 3 x 2 h + 3 x h 2 + h 3 − x 3 h Notice that the first term in the expansion of ( x + h ) 3 is x 3 and the second term is 3 x 2 h. For this function, both f ( x ) = c f ( x ) = c and f ( x + h ) = c, f ( x + h ) = c, so we obtain the following result:ĭ d x ( x 3 ) = lim h → 0 ( x + h ) 3 − x 3 h = lim h → 0 x 3 + 3 x 2 h + 3 x h 2 + h 3 − x 3 h Notice that the first term in the expansion of ( x + h ) 3 is x 3 and the second term is 3 x 2 h. We first apply the limit definition of the derivative to find the derivative of the constant function, f ( x ) = c. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions. The functions f ( x ) = c f ( x ) = c and g ( x ) = x n g ( x ) = x n where n n is a positive integer are the building blocks from which all polynomials and rational functions are constructed. In this section, we develop rules for finding derivatives that allow us to bypass this process. The process that we could use to evaluate d d x ( x 3 ) d d x ( x 3 ) using the definition, while similar, is more complicated. For example, previously we found that d d x ( x ) = 1 2 x d d x ( x ) = 1 2 x by using a process that involved multiplying an expression by a conjugate prior to evaluating a limit.

3.3.6 Combine the differentiation rules to find the derivative of a polynomial or rational function.įinding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process.

3.3.5 Extend the power rule to functions with negative exponents.

3.3.4 Use the quotient rule for finding the derivative of a quotient of functions.

3.3.3 Use the product rule for finding the derivative of a product of functions.

3.3.2 Apply the sum and difference rules to combine derivatives.

3.3.1 State the constant, constant multiple, and power rules.