미분 전체 문제

Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes.

$y=x-\sqrt{4-{x}^{2}}$

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 $K(t)=\left(\frac{1}{2}\right)m(t)v{(t)}^{2},$ 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, $\theta ?$

T/F: Let $f$ be a position function. The average rate of change on $[a,b]$ is the slope of the line through the points $(a, f(a))$ and $(b,f(b))$.

T/F: The definition of the derivative of a function at a point involves taking a limit.

In your own words, explain the difference between the average rate of change and instantaneous rate of change.

Let $y=f(x)$. Give three different notations equivalent to “$f\,'(x)$.”

If two lines are perpendicular, what is true of their slopes?

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle f(x) = 6$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle f(x) = 2x$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle f(t) = 4-3t$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle g(x) = x^2$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle h(x) = x^3$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle f(x) = 3x^2-x+4$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle r(x) = \frac1x$

In Exercises 7–14, use the definition of the derivative to compute the derivative of the given function.

$\displaystyle r(s) = \frac1{s-2}$

In Exercises 23–26, a function $f$ and an $x$–value $a$ are given. Approximate the equation of the tangent line to the graph of $f$ at $x=a$ by numerically approximating $f\,'(a)$, using $h=0.1$.

$f(x) = x^2+2x+1$, $x=3$

In Exercises 23–26, a function $f$ and an $x$–value $a$ are given. Approximate the equation of the tangent line to the graph of $f$ at $x=a$ by numerically approximating $f\,'(a)$, using $h=0.1$.

$\displaystyle f(x) = \frac{10}{x+1}$, $x=9$

In Exercises 23–26, a function $f$ and an $x$–value $a$ are given. Approximate the equation of the tangent line to the graph of $f$ at $x=a$ by numerically approximating $f\,'(a)$, using $h=0.1$.

$\displaystyle f(x) = e^x$, $x=2$