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3 years ago

The inverse of an exponential function is the logarithmic function. True or False?

1 answer:

5 0

For one-to-one functions, the inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function.
For example:
$f(x)=e^{x}$
$f^{-1} (x)=ln\ x$

$f(x)=ln\ x$
$f^{-1} x=e^{x}$

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How many terms of the arithmetic sequence {1,22,43,64,85,…} will give a sum of 2332? Show all steps including the formulas used

MA_775_DIABLO [31]

There's a slight problem with your question, but we'll get to that...

Consecutive terms of the sequence are separated by a fixed difference of 21 (22 = 1 + 21, 43 = 22 + 21, 64 = 43 + 21, and so on), so the n-th term of the sequence, a (n), is given recursively by

• a (1) = 1

• a (n) = a (n - 1) + 21 … … … for n > 1

We can find the explicit rule for the sequence by iterative substitution:

a (2) = a (1) + 21

a (3) = a (2) + 21 = (a (1) + 21) + 21 = a (1) + 2×21

a (4) = a (3) + 21 = (a (1) + 2×21) + 21 = a (1) + 3×21

and so on, with the general pattern

a (n) = a (1) + 21 (n - 1) = 21n - 20

Now, we're told that the sum of some number N of terms in this sequence is 2332. In other words, the N-th partial sum of the sequence is

a (1) + a (2) + a (3) + … + a (N - 1) + a (N) = 2332

or more compactly,

$\displaystyle\sum_{n=1}^N a(n) = 2332$

It's important to note that N must be some positive integer.

Replace a (n) by the explicit rule:

$\displaystyle\sum_{n=1}^N (21n-20) = 2332$

Expand the sum on the left as

$\displaystyle 21 \sum_{n=1}^N n-20\sum_{n=1}^N1 = 2332$

and recall the formulas,

$\displaystyle\sum_{k=1}^n1=\underbrace{1+1+\cdots+1}_{n\text{ times}}=n$

$\displaystyle\sum_{k=1}^nk=1+2+3+\cdots+n=\frac{n(n+1)}2$

So the sum of the first N terms of a (n) is such that

21 × N (N + 1)/2 - 20N = 2332

Solve for N :

21 (N ² + N) - 40N = 4664

21 N ² - 19 N - 4664 = 0

Now for the problem I mentioned at the start: this polynomial has no rational roots, and instead

N = (19 ± √392,137)/42 ≈ -14.45 or 15.36

so there is no positive integer N for which the first N terms of the sum add up to 2332.

4 0

2 years ago

IMPORTANT HELP HURRY

Zanzabum

Let Bea = x
Therefore Andrew = x + 2
Chris = x - 3

Andrew * Chris = 66
(x + 2)(x - 3) = 66 Remove the brackets.
x^2 + 2x - 3x - 6 = 66
x^2 - x - 6 = 66 Subtract 66 from both sides.
x^2 - x - 72 = 0
(x - 9)(x + 8) = 0
Find the possible values of Bea's age

x - 9 = 0
x = 9 That value works. She can be plus 9 years old.

x + 8 = 0
x = - 8. There's no way she can be - 8 years old

So Bea is 9.
9 <<<<<< Answer

6 0

3 years ago

If 3x+y = 7, what is the value of x/y? SOMEONE HELP I NEED IT QUICK!! plsss

andrey2020 [161]

Answer:

x=2 y=1

Step-by-step explanation:

3 times 2 =6

plus 1 = 7

8 0

3 years ago

-4+2(12) ———— -3(-3)-(-1)

dsp73

Answer:

20/10 is 2

final answer is 2

8 0

3 years ago

Describe the key features of the graph of the quadratic function f(x) = x2 - 8x - 20

Zolol [24]

If f(x)=x^2-8x-20 then

dy/dx=2x-8 and d2y/dx2=2

A) Since acceleration is a constant positive the parabola opens upward.

B) Since acceleration is a constant positive the parabola will have a vertex at an absolute minimum

C) The vertex will occur when dy/dx=0, 2x-8=0, 2x=8, x=4

So the axis of symmetry is the vertical line x=4.

The vertex is y(4)=-36, corresponding to the vertex point (4,-36)

The y-intercept occurs when x=0, y(0)=-20, so the y-intercept is the point (0,-20)

7 0

3 years ago