第4章微分の応用

A car is being driven at a rate of $40$ mph when the brakes are applied. The car decelerates at a constant rate of $10$ ft/sec². How long before the car stops?

In the preceding problem, calculate how far the car travels in the time it takes to stop.

You are merging onto the freeway, accelerating from rest at a constant rate of $12$ ft/sec². How long does it take you to reach merging speed at $60$ mph?

Based on the previous problem, how far does the car travel to reach merging speed?

A car company wants to ensure its newest model can stop in $8$ sec when traveling at $75$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

A car company wants to ensure its newest model can stop in less than $450$ ft when traveling at $60$ mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.

[T] $f\left(x\right)=x^2+2$

[T] $f\left(x\right)=4x-\sqrt{x}$

[T] $f\left(x\right)=\text{sin}x+2x$

[T] $f\left(x\right)=e^x$

[T] $f\left(x\right)=\frac{1}{{\left(x+1\right)}^2}$

[T] $f\left(x\right)=e^{\mathrm{-2}x}+3x^2$

If $f\left(x\right)$ is the antiderivative of $v\left(x\right),$ then $2f\left(x\right)$ is the antiderivative of $2v\left(x\right).$

If $f\left(x\right)$ is the antiderivative of $v\left(x\right),$ then $f\left(2x\right)$ is the antiderivative of $v\left(2x\right).$

If $f\left(x\right)$ is the antiderivative of $v\left(x\right),$ then $f\left(x\right)+1$ is the antiderivative of $v\left(x\right)+1.$

If $f\left(x\right)$ is the antiderivative of $v\left(x\right),$ then ${\left(f\left(x\right)\right)}^2$ is the antiderivative of ${\left(v\left(x\right)\right)}^2.$

If $f\left(\mathrm{-1}\right)=\mathrm{-6}$ and $f\left(1\right)=2,$ then there exists at least one point $x\in \left[\mathrm{-1},1\right]$ such that $f^{\prime }\left(x\right)=4.$

If $f^{\prime }\left(c\right)=0,$ there is a maximum or minimum at $x=c.$

There is a function such that $f\left(x\right)<0,f^{\prime }\left(x\right)>0,$ and $f^{″}\left(x\right)<0.$ (A graphical “proof” is acceptable for this answer.)

There is a function such that there is both an inflection point and a critical point for some value $x=a.$

Given the graph of $f^{\prime },$ determine where $f$ is increasing or decreasing.

The graph of $f$ is given below. Draw $f^{\prime }.$

Find the linear approximation $L\left(x\right)$ to $y=x^2+\text{tan}\left(\pi x\right)$ near $x=\frac{1}{4}.$

Find the differential of $y=x^2-5x-6$ and evaluate for $x=2$ with $dx=0.1.$

$f\left(x\right)=x+{\text{sin}}^2\left(x\right)$ over $\left[0,\pi \right]$

$f\left(x\right)=3x^4-4x^3-12x^2+6$ over $\left[\mathrm{-3},3\right]$

A car is being compacted into a rectangular solid. The volume is decreasing at a rate of $2$ m³/sec. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of $0.25$ m/sec, find the rate at which the height is changing when the length and width are $2$ m and the height is $1.5$ m.

A rocket is launched into space; its kinetic energy is given by $K\left(t\right)=\left(\frac{1}{2}\right)m\left(t\right)v{\left(t\right)}^2,$ where $K$ is the kinetic energy in joules, $m$ is the mass of the rocket in kilograms, and $v$ is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of $15$ m/sec² and the mass is decreasing at a rate of $10$ kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is $2000$ kg and the velocity is $5000$ m/sec? Give your answer in mega-Joules per second (MJ/s), which is equivalent to ${10}^6$ J/s.

The famous Regiomontanus’ problem for angle maximization was proposed during the 15th century. A painting hangs on a wall with the bottom of the painting a distance $a$ feet above eye level, and the top $b$ feet above eye level. What distance $x$ (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, $\theta ?$

An airline sells tickets from Tokyo to Detroit for $\text{\$}1200.$ There are $500$ seats available and a typical flight books $350$ seats. For every $\text{\$}10$ decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?