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3 years ago

Why must the base of an exponential function be positive?

Mathematics

1 answer:

ivolga24 [154]3 years ago

7 0

Answer:

Because we only work with positive bases, bx is always positive.

Step-by-step explanation:

The values of f(x), therefore, are either always positive or always negative, depending on the sign of a. Smaller values of b lead to faster rates of decay.

You might be interested in

Which point on the unit circle corresponds to −π ?

Arada [10]

On the unit circle:
( 1, 0 ) corresponds to 0.
( 0, 1 ) corresponds to π/2
( -1, 0 ) corresponds to π
( 0, - 1 ) corresponds to 3π/2
- π = π
Answer:
D ) ( - 1 , 0 )

6 0

2 years ago

Read 2 more answers

What is the area of the sector shown in the picture

tatuchka [14]

Answer:

The area of the sector shown is 4.75 pi

Step-by-step explanation:

First we have to know what fraction of the total circle that part represents

for that we divide 45 by 360

45/360 = 1/8

now we calculate the area of the circle and divide it by 8

a = ( π * 6² ) / 8

a = ( π * 36 ) / 8

a = π * 4.75

a = 4.75 pi

7 0

3 years ago

Cody is twice as old Evan and five years ago the sum of their age was 14. Find the age of each boy

Pani-rosa [81]

Cody is 15 and evan is 10.

6 0

2 years ago

How to find the derivative of cos^2x? i seem to be confused.

slamgirl [31]

If you're using the app, try seeing this answer through your browser: brainly.com/question/2927231

————————

You can actually use either the product rule or the chain rule for this one. Observe:

• Method I:

y = cos² x

y = cos x · cos x

Differentiate it by applying the product rule:

$\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(cos\,x\cdot cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(cos\,x)\cdot cos\,x+cos\,x\cdot \dfrac{d}{dx}(cos\,x)}$

The derivative of cos x is – sin x. So you have

$\mathsf{\dfrac{dy}{dx}=(-sin\,x)\cdot cos\,x+cos\,x\cdot (-sin\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=-sin\,x\cdot cos\,x-cos\,x\cdot sin\,x}$

$\therefore~~\boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=-2\,sin\,x\cdot cos\,x}\end{array}}\qquad\quad\checkmark$

—————

• Method II:

You can also treat y as a composite function:

$\left\{\!
\begin{array}{l}
\mathsf{y=u^2}\\\\
\mathsf{u=cos\,x}
\end{array}
\right.$

and then, differentiate y by applying the chain rule:

$\mathsf{\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}}\\\\\\
\mathsf{\dfrac{dy}{dx}=\dfrac{d}{du}(u^2)\cdot \dfrac{d}{dx}(cos\,x)}$

For that first derivative with respect to u, just use the power rule, then you have

$\mathsf{\dfrac{dy}{dx}=2u^{2-1}\cdot \dfrac{d}{dx}(cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=2u\cdot (-sin\,x)\qquad\quad (but~~u=cos\,x)}\\\\\\
\mathsf{\dfrac{dy}{dx}=2\,cos\,x\cdot (-sin\,x)}$

and then you get the same answer:

$\therefore~~\boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=-2\,sin\,x\cdot cos\,x}\end{array}}\qquad\quad\checkmark$

I hope this helps. =)

Tags: <em>derivative chain rule product rule composite function trigonometric trig squared cosine cos differential integral calculus</em>

3 0

3 years ago

Jacob decided to ride his bicycle across the country during his 2-month summer vacation. the route he took from washington, dc t

Artist 52 [7]

Step 1: Assign variable for the unknown that we need to find.

Let ' x ' be the number of miles traveled by Jacob so far (His distance from starting point)

Step 2: Use the sentence given to set up an equation

Sentence 1: "His distance from his starting point was exactly 100 miles more than three times the distance remaining until the finishing point"

Total distance is 2800 miles, if his distance from the starting point is 'x', then remaining distance can be represented by 2800 - x.

Using the sentence we can write the below equation...

$x = 100 + 3(2800 - x)$