Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. A payload is dropped at an elevation of $400 \mathrm{~m}$ from a hot-air balloon that is descending at a rate of $10 \mathrm{~m} / \mathrm{s}$. Find the time when the object reaches its highest point. What is the height?

What two nonnegative real numbers with a sum of 23 have the largest possible product?

An object moves along a line with a velocity in m/s given by $v(t)=8 \cos \frac{\pi t}{6}$. Its initial position is $s(0)=0$. As discussed in Chapter 6 , the position of the object is given by $s(t)=\int_0^t v(y) d y$, for $t \geq 0$. Find the position function, for $t \geq 0$.

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation $a(t)=v^{\prime}(t)=-g$, where $g=9.8 \mathrm{m} / \mathrm{s}^{2}$. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A payload is dropped at an elevation of 400 1n from a hot-air balloon that is descending at a rate of 10 m/s.

Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. f(x)=

$$\frac { \sqrt { 4 x ^ { 2 } + 1 } } { x ^ { 2 } + 1 }$$

An object moves along a line with a velocity in m/s given by $v(t)=8 \cos \frac{\pi t}{6}$. Its initial position is $s(0)=0$. Graph the velocity function.

Simplify the given expressions. $\frac { d } { d x } \int _ { 0 } ^ { x ^ { 2 } } \frac { d t } { t ^ { 2 } + 4 }$

Determine the following indefinite integrals.

$\int(3 x+1)(4-x) d x$

Sketch the graph of a function f that has a local minimum value at a point c where f'(c) is undefined..

Explain why

$$\int _ { a } ^ { b }$$

f'(x)dx=f(b)-f(a).

Find the function f that satisfies $f^{\prime}(t)=e^t-2 t+3$ and $f(0)=4$

(A) $f(t)=e^t-t^2+3$

(B) $f(t)=2 e^t-t^2+3 t+2$

(C) $f(t)=e^t-t^2+3 t+3$

(D) $f(t)=e^t-\frac{1}{2} t^2+3 t+3$

Compute the absolute and relative errors in using c to approximate x. x=e; c=2.718

Determine the following indefinite integrals. Check your work by differentiation.

$$\int \sec 4 \theta \tan 4 \theta d \theta$$

Give a geometrical explanation of why

$$\int _ { a } ^ { a } f ( x ) d x = 0$$

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