MATHEMATICS COLLEGE

James has $20.00 in his checking account. He goes to the bank and withdraws $20.00. How much money does James have in his account immediately after withdrawing the $20.00?

Answers

Answer 1

Answer:

Answer:

$0.00

Step-by-step explanation:

$20.00-$20.00=$0.00

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Find the limit of the formula given

Answers

Answer:

$\displaystyle \lim_(x \to 0^+) x^\big{√(x)} = 1$

General Formulas and Concepts:

Algebra II

Natural logarithms ln and Euler's number e
Logarithmic Property [Exponential]: $\displaystyle log(a^b) = b \cdot log(a)$

Calculus

Limits

Right-Side Limit: $\displaystyle \lim_(x \to c^+) f(x)$
Left-Side Limit: $\displaystyle \lim_(x \to c^-) f(x)$

Limit Rule [Variable Direct Substitution]: $\displaystyle \lim_(x \to c) x = c$

L’Hopital’s Rule: $\displaystyle \lim_(x \to c) (f(x))/(g(x)) = \lim_(x \to c) (f'(x))/(g'(x))$

Differentiation

Derivatives
Derivative Notation

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

We are given the following limit:

$\displaystyle \lim_(x \to 0^+) x^\big{√(x)}$

Substituting in x = 0 using the limit rule, we have an indeterminate form:

$\displaystyle \lim_(x \to 0^+) x^\big{√(x)} = 0^0$

We need to rewrite this indeterminate form to another form to use L'Hopital's Rule. Let's set our limit as a function:

$\displaystyle y = \lim_(x \to 0^+) x^\big{√(x)}$

Take the ln of both sides:

$\displaystyle lny = ln \Big( \lim_(x \to 0^+) x^\big{√(x)} \Big)$

Rewrite the limit by including the ln in the inside:

$\displaystyle lny = \lim_(x \to 0^+) ln \big( x^\big{√(x)} \big)$

Rewrite the limit once more using logarithmic properties:

$\displaystyle lny = \lim_(x \to 0^+) √(x)ln(x)$

Rewrite the limit again:

$\displaystyle lny = \lim_(x \to 0^+) (ln(x))/((1)/(√(x)))$

Substitute in x = 0 again using the limit rule, we have an indeterminate form in which we can use L'Hopital's Rule:

$\displaystyle \lim_(x \to 0^+) (ln(x))/((1)/(√(x))) = (\infty)/(\infty)$

Apply L'Hopital's Rule:

$\displaystyle \lim_(x \to 0^+) (ln(x))/((1)/(√(x))) = \lim_(x \to 0^+) \frac{(1)/(x)}{\frac{-1}{2x^\big{(3)/(2)}}}$

Simplify:

$\displaystyle \lim_(x \to 0^+) \frac{(1)/(x)}{\frac{-1}{2x^\big{(3)/(2)}}} = \lim_(x \to 0^+) -2√(x)$

Redefine the limit:

$\displaystyle lny = \lim_(x \to 0^+) -2√(x)$

Substitute in x = 0 once more using the limit rule:

$\displaystyle \lim_(x \to 0^+) -2√(x) = -2√(0)$

Evaluating it, we have:

$\displaystyle \lim_(x \to 0^+) -2√(x) = 0$

Substitute in the limit value:

$\displaystyle lny = 0$

e both sides:

$\displaystyle e^\big{lny} = e^\big{0}$

Simplify:

$\displaystyle y = 1$

And we have our final answer.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

A trapezoid has an area of 60 square inches. The height of the trapezoid is 5 inches. What is the length of the longer base if the longer base is three times the length of the shorter base?

Answers

Answer:

Step-by-step explanation:

Remark

Let the shorter base = x

Let the longer base = 3x

h = 5

Area = 60

Formula

Area = (b1 + b2)*h /2

Solution

60 = (x + 3x)*5 / 2 Multiply both sides by 2

2*60 = (x + 3x)*5 Combine like terms

120 = 4x *5

120 = 20x Divide by 20

120/20 = x

x = 6

Therefore the two bases are

x = 6

3x = 18

Answer:

Step-by-step explanation:

Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of μ = 8.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words is σ = 2.5. Now a manuscript with six new sonnets has come to light, and scholars are debating whether it is the poet’s work. The new sonnets contain an average of x~ = 10.2 words not used in the poet’s known works. We expect poems by another author to contain more new words, so to see if we have evidence that the new sonnets are not by our poet we test the following hypotheses.H0 : µ = 8.88 vs Ha : µ > 8.88
Give the z test statistic and its P-value. What do you conclude about the authorship of the new poems? (Let a = .05.)
Use 2 decimal places for the z-score and 4 for the p-value.
a. What is z?
b.The p-value is greater than?
c.What is the conclusion? A)The sonnets were written by another poet or b) There is not enough evidence to reject the null.

Answers

Answer:

We conclude that the sonnets were written by by a certain Elizabethan poet.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 8.9

Sample mean, $\bar{x}$ =10.2

Sample size, n = 6

Alpha, α = 0.05

Population standard deviation, σ = 2.5

First, we design the null and the alternate hypothesis

$H_(0): \mu = 8.88\nH_A: \mu > 8.88$

We use One-tailed z test to perform this hypothesis.

a) Formula:

$z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }$

Putting all the values, we have

$z_(stat) = \displaystyle(10.2 - 8.9)/((2.5)/(√(6)) ) = 1.28$

Now, $z_(critical) \text{ at 0.05 level of significance } = 1.64$

b) We calculate the p value with the help of z-table.

P-value = 0.1003

The p-value is greater than the significance level which is 0.05

c) Since the p-value is greater than the significance level, there is not enough evidence to reject the null hypothesis and accept the null hypothesis.

Thus, we conclude that the sonnets were written by by a certain Elizabethan poet.

Final answer:

The z-score is 1.86 and the p-value is 0.0314. As the p-value is less than the level of significance α (0.05), we reject the null hypothesis and conclude that the new sonnets were likely written by another author.

Explanation:

In this statistical testing scenario for authorship of literary works, we need to find out the z-score or z test statistic and then determine the p-value to check if the new sonnets could be the works of the known Elizabethan poet or not.

For calculating the z score, you use the formula z = (x~ - μ) / (σ / √n) = (10.2 - 8.9) / (2.5/ √6) = 1.86 to two decimal places. The p-value is determined from the standard normal distribution table which for a z-score of 1.86 is 0.0314.

Given that α = 0.05, since the p-value is less than α, we reject the null hypothesis H0 (that the works were by the Elizabethan poet). Therefore, we accept the alternative hypothesis Ha (the sonnets were written by another author).

Learn more about Statistical Testing for Authorship here:

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Find all solutions in the interval [0, 2π).

4 sin^2 x - 4 sin x + 1 = 0

Answers

Answer:

$the \: solutions \: are : \n x = 0 \n x = 360$

Is the Mean Value Theorem applicable to the function f(x) = |x - 1| on the interval [0, 2]?Why or why not?

Answers

The only point that derivative of the function f(x) = |x - 1| is not continuous is at x = 0. You need to check whether the slope for the interval (0,2) is continuous to see if you can apply MVT. The interval (0,2) does not include end points, so 0 is not in this interval. The function is continuous over the interval, so MVT can be applied.

Question 8 of 10 What is the solution to the following equation? X2 + 5x + 7 = 0

Answers

Answer:

Step-by-step explanation:

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