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evablogger [386]
3 years ago
15

Consider the error in using the approximation e−x≈1−x on the interval [−1,1]. (a) reasoning informally, on what interval is this

approximation an overestimate? 1 an underestimate? (for each, give your answer as an interval or list of intervals,
e.g., to specify the intervals −0.25≤x<0.5 and 0.75
Mathematics
1 answer:
crimeas [40]3 years ago
6 0

e^{-x} \approx 1 - x

is a classic approximation, true for small x.

The next term in the polynomial expansion will be +kx^2 where k is a positive number. So our estimate 1-x is definitely an underestimate on both sides of x=0.

Since for negative xs the exponential rises exponentially and the line only linearly, the exponential exceeds the line for all negative x. For positive x, the line quickly goes negative while the exponential is always positive.

So, there's no interval for which our approximation is an overestimate.

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2 years ago
What is 3 2/7 x 2 4/5 (Answer with a mixed number in simplest form) Thankyou!!
SCORPION-xisa [38]

Answer:

9 1/5

Step-by-step explanation:

3 2/7 * 2 4/5

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7 0
3 years ago
Read 2 more answers
Simplify the following expression <br> (X9 Y5) -4
KATRIN_1 [288]

Answer:

Step-by-step explanation:

( x^{9} y^{5} )^{-4} \\x^{(9)(-4)} y^{(5)(-4)} \\x^{-36}y^{-20} \\\frac{1}{x^{36}y^{20}}

3 0
2 years ago
Read 2 more answers
F(x) = <img src="https://tex.z-dn.net/?f=%5Csqrt%7Bx%2B7%7D%20-%5Csqrt%7Bx%5E2%2B2x-15%7D" id="TexFormula1" title="\sqrt{x+7} -\
elena-s [515]

Answer:

x >= -7  ................(1a)

x >= 3   ...............(2a1)

Step-by-step explanation:

f(x) =  \sqrt{x+7}-\sqrt{x^2+2x-15}  .............(0)

find the domain.

To find the (real) domain, we need to ensure that each term remains a real number.

which means the following conditions must be met

x+7 >= 0  .....................(1)

AND

x^2+2x-15 >= 0 ..........(2)

To satisfy (1),  x >= -7  .....................(1a) by transposition of (1)

To satisfy (2), we need first to find the roots of (2)

factor (2)

(x+5)(x-3) >= 0

This implis

(x+5) >= 0 AND (x-3) >= 0.....................(2a)

OR

(x+5) <= 0 AND (x-3) <= 0 ...................(2b)

(2a) is satisfied with x >= 3   ...............(2a1)

(2b) is satisfied with x <= -5 ................(2b1)

Combine the conditions (1a), (2a1), and (2b1),

x >= -7  ................(1a)

AND

(

x >= 3   ...............(2a1)

OR

x <= -5 ................(2b1)

)

which expands to

(1a) and (2a1)   OR  (1a) and (2b1)

( x >= -7 and x >= 3 )  OR ( x >= -7 and x <= -5 )

Simplifying, we have

x >= 3  OR ( -7 <= x <= -5 )

Referring to attached figure, the domain is indicated in dark (purple), the red-brown and white regions satisfiy only one of the two conditions.

3 0
3 years ago
Read 2 more answers
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