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
what is 15% of 22 help
Hi please help me with this asap.
Use differentials to estimate the amount of material in a closed cylindrical can that is 60 cm high and 24 cm in diameter if the metal in the top and bottom is 0.1 cm thick, and the metal in the sides is 0.05 cm thick. Note, you are approximating the volume of metal which makes up the can (i.e. melt the can into a blob and measure its volume), not the volume it encloses. The differential for the volume is
Answer:
dV ≈ 927.4 cm³
Step-by-step explanation:
Formula for volume of a cylinder is;
V = πr²h
Using differentials, we have;
dV = (dV/dr)dr + (dV/dh)dh
Now, dV/dr = 2πrh
dV/dh = πr²
Thus, at r = 24/2 = 12 cm
And at h = 60 cm, we have;
dV/dr = 2π(12 × 60)
dV/dr = 4523.89 cm²
Also;
dV/dh = π(12)²
dV/dh = 452.39 cm²
the metal in the top and bottom is 0.1 cm thick. Thus, dr = 2 × 0.1 = 0.2
the metal in the sides is 0.05 cm thick. Thus; dh = 0.05
Thus;
dV = (4523.89 × 0.2) + (452.39 × 0.05)
dV ≈ 927.4 cm³
The ———— is used to determine the left-hand and right-hand behavior of the graph of a polynomial function.
--Select-
Behavior Test
t Answ Polynomial Test
Odd Degree Test
Horizontal Line Test
Leading Coefficient Test
Dints] Distinct Zero Test
Even Degree Test
Vertical Line Test
AT10 3.2.001.
Answer:
I think it is the vertical line test. If you have any other questions let me know! :)
Given y = alog(x + c) + d. What would happen to the graph if you changed the value of d? Describe both positive and negative
values.
A)
B)
Changing the value of d would cause a vertical stretch or shrink. A negative
value would also cause a reflection over the x-axis.
Changing the value of d would cause a horizontal stretch or shrink. A
negative value would also cause a reflection over the y-axis.
Changing the value of d would move the graph up and down (vertically). A
positive d value would move the graph up and a negative d value would
move the graph down.
Changing the value of d would move the graph left and right (horizontally).
A positive d value would move the graph right, and a negative d value
would move the graph left.
C)
D)
Answer: A
Step-by-step explanation:
I just took the USATESTPREP quiz and that is the answer.
Option (C) will define the changes in the graph of the function with the values of 'd'.
Given function in the question → y = alog(x + c) + d
Rules for the translation of a function,
If a function y = f(x) is shifted 'h' units upwards on y-axis, rule for the translation will be → y = f(x) + hIf the function y = f(x) is shifted 'h' units downwards, rule for the translation → y = f(x) - hHence, in the given function y = alog(x + c) + d, positive value of 'd' will define the shifting of the parent function upwards.
And for the negative value of 'd' will define the shift of the function downwards.
Option (C) will be the correct option.
Learn more,
https://brainly.com/question/16770247
Emily surveyed 364 people with 3 out of 7 answered yes
levi was designing a kite. he needs to determine the measure of
The measure of angle A is 86.
What is a kite?A kite is a flying object that is tethered to the ground by a string or a cord. It typically has a light framework of bamboo, plastic, or other materials, covered with paper or fabric, and often in the shape of a diamond or triangle.
We have,
From the figure,
We see that,
6x + 4 + ∠A = 180
Now,
∠A = 4x + 26
So,
6x + 4 + 4x + 26 = 180
10x + 30 = 180
10x = 180 - 30
10x = 150
x = 15
Now,
∠A = 4x + 26 = 4 x 15 + 26 = 60 + 26 = 86
Thus,
The measure of angle A is 86.
Learn more about kite here:
https://brainly.com/question/31239785
#SPJ3
The complete question:
Levi was designing a kite. he needs to determine the measure of angle A.
Six trigonometric functions of -315.
If the volume of a sphere is 660 mm, what is the radius of the sphere, to the nearest tenth of a millimeter?
Answer:
Radius= 5.4
Step-by-step explanation:
Formula for Sphere- V= 4/3(pi)• r^3
r=(3 V/4 pi) = (3•660/4•pi)= 5.40114 (5.4)
20 baseballs pitched he hit 17. What %did TJ hit
Find the volume of the cylinder.
Either enter an exact answer in terms of or use 3.14 for pi
Answer:
Answer in explanation
Step-by-step explanation:
Volume of cylinder =
[tex]\pi {r}^{2} h \\ = (3.14)(4) {}^{2}(8) \\ = 401.92units {}^{3} \: or \: 128\pi \: units^{3} [/tex]
Answer:
Step-by-step explanation:
Radius r = 4
Height h = 8
Volume = πr²h = 128π units³
What are the ordered pairs of the
solutions for this system of equations?
f(x) = x2 – 2x + 3; f(x) = -2x + 12
Answer:
Step-by-step explanation:
Find the value of x
Answer:
x=13
Step-by-step explanation:
Since both angles are vertical angles and are therefore congruent, the expression is equal to 114.
7x+23=114
Subtract 23 from both sides.
7x=91
Divide both sides by 7.
x=13
Someone please help me, i will give brainliest
Enter the decimal equivalent of 11/8
Answer:
1.37
Step-by-step explanation:
I just used my calculator lol
(1 point)
10. A piece of art is in the shape of an equilateral triangle with sides of 21 in. What is the area of
the piece of art to the nearest tenth?
311.8 in.2
381.9 in?
155.9 in.2
191.0 in.2
Answer:
191.0
Step-by-step explanation:
The perimeter of the shape below is 44 feet. What is the area in square feet?
Answer:
2(4x+2+6x)=44
10x+2=22
10x=20
x=2
area= 10*12=120sq feet
Step-by-step explanation:
Please help me ASAP
Please help
Answer:
5 X 4
Step-by-step explanation:
If you plot the dots, youll obviously see a rectangle. The height of the rect. is 4, while the length is 5, so the measurements would be 5 x 4.
Hope this helped!!
simplify [tex]\sqrt{p^4q^6r^5. answer is D just took the test
Helpppp!!!! ASAP!! Please???!
Answer:
c 111 mark me
Step-by-step explanation:
and ur welcome lol
Answer:
105
Step-by-step explanation:
if it true i add 65 +40 =105 so yeah
How many 6-digit numbers are there in our system of counting numbers?
there are 900,000 6 in all
brainliest or a thank you please <33 :)
The following is the estimation results for a multiple linear regression model:
Y = B. + ,X,+B,X, +E
SUMMARY OUTPUT
Regression Statistics
R-Square 0.558
Regression Standard Error (S) 863.100
Observations 35
Coeff StdError t-stat
Intercept 1283.000 352.000 3.65
X1 25.228 8.631
X2 0.861 0.372
Questions:
1. Write the fitted regression equation (30 points).
2. Write the estimated intercepts and slopes, associated with their corresponding standard errors (30 points).
3. Interpret each coefficient (40 points).
Answer:
Kindly check explanation
Step-by-step explanation:
Given the summary output :
_______ Coeff___ StdError__ t-stat
Intercept 1283.000_ 352.000 _ 3.65
X1 _____ 25.228 ____8.631
X2 _____0.861 _____0.372
General form of a multiple regression model:
Y = B1X1+B2X2 +E
The fitted regression equation is :
y = 25.228X1 + 0.861X2 + 1283
Intercept and slope corresponding to the standard error :
Intercept = 352
Slope 1 = 8.61 ; slope 2 = 0.372
'B' coefficient represents the change in the mean response, Y, per unit increase in a version predictor variable when all the other predictors are held constant.
B1 represents the change in the mean response, U, per unit increase in x1 when x2, x3, ..., xk are held constant
B2 represents the change in the mean response, U, per unit increase in x2 when x1, x3, ..., xk are held constant
Describe the graph of y=-x and compare it with the graph of y = x
I need this thank you please help
Answer:
Step-by-step explanation:
This is not a function.
A function has to have 1 y associated with each x. This relationship has 2 except for 1 point (the vertex at 3,0).
You can write a domain and range for it.
The domain is
- infinity < x < 3
The range is
-infinity < x < + infinity.
Which angle is congruent to B
I NEED HELP QUICK!!!
Answer: 12cm
Step-by-step explanation:
What is the scale factor from ABC to DEF
Answer:
Since you are looking for the scale factor of ABC to DEF the answer is 8 because DEF is 8 times larger than ABC
Step-by-step explanation:
Given f(x)=x^2−5x and g(x)=2x+1, find (f+g)(−3).
Answer:
19
Use the given functions to set up and simplify
I need help with Algebra 2. Thank you! :)
Answer:
B: 6.0
Step-by-step explanation:
-3t^2+9t+54 = 0
This is the equation that needs to be solved
Divide both sides by -3
t^2 - 3t - 18 = 0
The factors of 18 are: (1 18) (2 9) (3 6)
3 - 6 = -3 and 3 x -6 = -18
So: (t + 3) (t - 6) = 0
t = -3, 6
It can't be negative so the answer is 6
can someone help me please
Answer:
Step-by-step explanation:
Answer:
The correct answer is B. $6.50 per pound.
3. A pharmacist needs a blend of chemical containing 35% of manganese but she has one vial
at 40% and another at 25%. How many of each must be mixed to provide 100 pounds of
35% manganese?