The position of an object on a coordinate axis at time
$t$ is given by
$s\left(t\right)=\frac{t}{{t}^{2}+1}.$ What is the initial velocity of the object?
Since the initial velocity is
$v\left(0\right)={s}^{\prime}\left(0\right),$ begin by finding
${s}^{\prime}(t)$ by applying the quotient rule:
Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (
[link] ).
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Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. 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.
Suppose you are designing a new Formula One track. One section of the track can be modeled by the function
$f\left(x\right)={x}^{3}+3x+x$ (
[link] ). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. The plans call for the front corner of the grandstand to be located at the point
$\left(\mathrm{-1.9},2.8\right).$ We want to determine whether this location puts the spectators in danger if a driver loses control of the car.
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. Find the
$\left(x,y\right)$ coordinates of this point near the turn.
Find the equation of the tangent line to the curve at this point.
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
$y=2.8.$ Is this point safely to the right of the grandstand? Or are the spectators in danger?
What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point
$\left(\mathrm{-2.5},0.625\right).$ What is the slope of the tangent line at this point?
If a driver loses control as described in part 4, are the spectators safe?
Should you proceed with the current design for the grandstand, or should the grandstands be moved?
Key concepts
The derivative of a constant function is zero.
The derivative of a power function is a function in which the power on
$x$ becomes the coefficient of the term and the power on
$x$ in the derivative decreases by 1.
The derivative of a constant
c multiplied by a function
f is the same as the constant multiplied by the derivative.
The derivative of the sum of a function
f and a function
g is the same as the sum of the derivative of
f and the derivative of
g.
The derivative of the difference of a function
f and a function
g is the same as the difference of the derivative of
f and the derivative of
g.
The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |.
The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.