What is the minimum number of points required to make a plane

an exponential function f is defined by f(x)=c^x where c is a constant greater than 1 if f (7) = 4 x f (5) what is the value of

svetoff [14.1K]

From the above, it can be seen that the nature of polynomial functions is dependent on its degree. Higher the degree of any polynomial function, then higher is its growth. A function which grows faster than a polynomial function is y = f(x) = ax, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of fn(x).

Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain of exponential function will be the set of entire real numbers R and the range are said to be the set of all the positive real numbers.

It must be noted that exponential function is increasing and the point (0, 1) always lies on the graph of an exponential function. Also, it is very close to zero if the value of x is mostly negative.

Exponential function having base 10 is known as a common exponential function. Consider the following series:

Derivative of logarithmic and exponential function 5

The value of this series lies between 2 & 3. It is represented by e. Keeping e as base the function, we get y = ex, which is a very important function in mathematics known as a natural exponential function.

For a > 1, the logarithm of b to base a is x if ax = b. Thus, loga b = x if ax = b. This function is known as logarithmic function.

Derivative of logarithmic and exponential function 2

For base a = 10, this function is known as common logarithm and for the base a = e, it is known as natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which have a base a>1.

The domain of log function consists of positive real numbers only, as we cannot interpret the meaning of log functions for negative values.

For the log function, though the domain is only the set of positive real numbers, the range is set of all real values, i.e. R

When we plot the graph of log functions and move from left to right, the functions show increasing behaviour.

The graph of log function never cuts x-axis or y-axis, though it seems to tend towards them.

Derivative of logarithmic and exponential function 3

Logap = α, logbp = β and logba = µ, then aα = p, bβ = p and bµ = a

Logbpq = Logbp + Logbq

Logbpy = ylogbp

Logb (p/q) = logbp – logbq

Exponential Function Derivative

Let us now focus on the derivative of exponential functions.

The derivative of ex with respect to x is ex, i.e. d(ex)/dx = ex

It is noted that the exponential function f(x) =ex has a special property. It means that the derivative of the function is the function itself.

(i.e) f ‘(x) = ex = f(x)

Exponential Series

Exponential Functions

Exponential Function Properties

The exponential graph of a function represents the exponential function properties.

Let us consider the exponential function, y=2x

The graph of function y=2x is shown below. First, the property of the exponential function graph when the base is greater than 1.

Exponential Functions

Exponential Function Graph for y=2x

The graph passes through the point (0,1).

The domain is all real numbers

The range is y>0

The graph is increasing

The graph is asymptotic to the x-axis as x approaches negative infinity

The graph increases without bound as x approaches positive infinity

The graph is continuous

The graph is smooth

Exponential Functions

Exponential Function Graph y=2-x

The graph of function y=2-x is shown above. The properties of the exponential function and its graph when the base is between 0 and 1 are given.

The line passes through the point (0,1)

The domain includes all real numbers

The range is of y>0

It forms a decreasing graph

The line in the graph above is asymptotic to the x-axis as x approaches positive infinity

The line increases without bound as x approaches negative infinity

It is a continuous graph

It forms a smooth graph

Exponential Function Rules

Some important exponential rules are given below:

If a>0, and b>0, the following hold true for all the real numbers x and y:

ax ay = ax+y

ax/ay = ax-y

(ax)y = axy

axbx=(ab)x

(a/b)x= ax/bx

a0=1

a-x= 1/ ax

Exponential Functions Examples

The examples of exponential functions are:

f(x) = 2x

f(x) = 1/ 2x = 2-x

f(x) = 2x+3

f(x) = 0.5x

Solved problem

Question:

Simplify the exponential equation 2x-2x+1

Solution:

Given exponential equation: 2x-2x+1

By using the property: ax ay = ax+y

Hence, 2x+1 can be written as 2x. 2

Thus the given equation is written as:

2x-2x+1 =2x-2x. 2

Now, factor out the term 2x

2x-2x+1 =2x-2x. 2 = 2x(1-2)

2x-2x+1 = 2x(-1)

2x-2x+1 = – 2x

6 1

3 years ago

Folds - Circle Cliffs, UT. Both monoclines and anticlines uplift and deform rock layers such that when eroded, certain ages of r

DIA [1.3K]

Answer:the relative ages of the rocks exposed in the circle cliff area are given below.

1. older rocks are exposed in the center and younger rocks in the flanking flatirons.

2. younger rocks are exposed in the center and older rocks in the flanking flatirons.

Step-by-step explanation: this is because whenever older rocks are exposed in circle cliffs, exposure occur at the center while younger rocks will be exposed in the flanking flatirons at this time.

2. But when younger rocks are exposed in contrast to the older rocks, these younger rocks are exposed at the center while the older ones receive exposure at the flanking flatirons.

Note that both cases interchange, exposure of a particular rock occur at the center and the next category of rock receive theirs at flanking flatirons.

7 0

3 years ago

Yuuma applied the steps below to find the product of (–10.2)(–16.4).

goblinko [34]

Answer:

164+3.28

Step-by-step explanation:

(16.4)(10)+(16.4)(0.2)

164+3.28

167.28

4 0

2 years ago

6. The population of a town is 680 000 correct to the nearest 10 000. Write down

serg [7]

Answer:

a) 675 000

b) 685 000

Step-by-step explanation:

The population of a town is 680 000 correct to the nearest 10 000.

a) To find it lower bound, we level of accuracy by 2 and then subtract from 680 000

The lower bound is:

680 000-5000=675,000

Therefore the least possible population of the town is 675 000

b) We repeat the same process to find the upper bound

680 000+5000=685,000

6 0

4 years ago

If sinA+cosecA=3 find the value of sin2A+cosec2A

Irina18 [472]

Answer:

$\sin 2A + \csc 2A = 2.122$

Step-by-step explanation:

Let $f(A) = \sin A + \csc A$ , we proceed to transform the expression into an equivalent form of sines and cosines by means of the following trigonometrical identity:

$\csc A = \frac{1}{\sin A}$ (1)

$\sin^{2}A +\cos^{2}A = 1$ (2)

Now we perform the operations: $f(A) = 3$

$\sin A + \csc A = 3$

$\sin A + \frac{1}{\sin A} = 3$

$\sin ^{2}A + 1 = 3\cdot \sin A$

$\sin^{2}A -3\cdot \sin A +1 = 0$ (3)

By the quadratic formula, we find the following solutions:

$\sin A_{1} \approx 2.618$ and $\sin A_{2} \approx 0.382$

Since sine is a bounded function between -1 and 1, the only solution that is mathematically reasonable is:

$\sin A \approx 0.382$

By means of inverse trigonometrical function, we get the value associate of the function in sexagesimal degrees:

$A \approx 22.457^{\circ}$

Then, the values of the cosine associated with that angle is:

$\cos A \approx 0.924$

Now, we have that $f(A) = \sin 2A +\csc2A$ , we proceed to transform the expression into an equivalent form with sines and cosines. The following trignometrical identities are used: