The area of the region bounded by the curve and the tangent line is approximately 2.197 square units.
To find the area of the region bounded by the curve and the tangent line, we need to find the x-coordinate where the tangent line is tangent to the curve. This can be found by setting the derivative of the curve equal to the slope of the tangent line at that point.
The derivative of the curve is:
f'(x) = 3x^2 - 4
Setting this equal to the slope of the tangent line, which is the derivative of the curve at the tangent point, we get:
f'(x) = 3x^2 - 4 = 3
Solving for x, we get:
x^2 = 3/3 = 1
x = ±1
We only need to consider the positive value of x, since the tangent line will be tangent to the curve at both x = 1 and x = -1.
At x = 1, the y-coordinate of the curve is:
f(1) = 1^3 - 4(1) + 1 = -2
The slope of the tangent line at x = 1 is:
f'(1) = 3(1)^2 - 4 = -1
So the equation of the tangent line at x = 1 is:
y + 2 = -1(x - 1)
Simplifying, we get:
y = -x + 1
To find the area of the region bounded by the curve and the tangent line, we need to find the x-coordinates where they intersect. Setting the equations equal to each other, we get:
x^3 - 4x + 1 = -x + 1
Simplifying, we get:
x^3 - 3x = 0
x(x^2 - 3) = 0
x = 0 or x = ±sqrt(3)
We only need to consider the positive value of x, since the tangent line intersects the curve at both x = sqrt(3) and x = -sqrt(3).
At x = sqrt(3), the y-coordinate of the curve is:
f(sqrt(3)) = (sqrt(3))^3 - 4(sqrt(3)) + 1 ≈ -0.732
At x = sqrt(3), the y-coordinate of the tangent line is:
y = -sqrt(3) + 1 ≈ -0.732
So the height of the region is approximately:
h ≈ |-0.732 - (-2)| = 1.268
The base of the region is:
b = sqrt(3)
So the area of the region is approximately:
A ≈ bh ≈ sqrt(3) * 1.268 ≈ 2.197
Therefore, the area of the region bounded by the curve and the tangent line is approximately 2.197 square units.
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How do the graphs of the functions f(x) = (3/2)x and g(x)
The function g(x) = (2/3)ˣ is the graph of the function f(x) = (3/2)ˣ reflected over y-axis.
What is function?A function is a process or a relation that associates each element 'a' of a non-empty set A , at least to a single element 'b' of another non-empty set B. A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math.
f = {(a, b)| for all a ∈ A, b ∈ B}.
Given, the function
f(x) = (3/2)ˣ
g(x) = (2/3)ˣ
We have to compare the graphs of the functions f(x) and g(x).
Exponential function is of the form: y = abˣ
Where a is the initial value and b is the growth factor.
a) If b > 1, then the function is an exponential growth function.
b) If 0 < b < 1, then the function is an exponential decay function.
f(x) = y = (3/2)ˣ
The function is an exponential growth function.
The rule of reflection over y-axis:
(x, y) → (-x, y)
Applying the rule, we get,
(3/2)ˣ = (3/2)⁻ˣ
= (2/3)ˣ
= g(x)
Therefore, g(x) is graph of f(x) reflected over y-axis
Hence, graph of the function f(x) = (3/2)ˣ reflected over y-axis is function g(x) = (2/3)ˣ .
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What is the additive inverse of 14 /- 15?
The additive inverse of 14 /- 15 is -14 /+ 15.
The antithesis of a number is the additive inverse. It is found by changing the sign of the number. To find the additive inverse of 14 /- 15, start by changing the sign of the number from a negative sign to a positive sign. The result is 14 /+ 15. This is the additive inverse of 14 /- 15. The additive inverse of any number is the same number, but with the opposite sign. For example, the additive inverse of -14 is 14, and the additive inverse of 14 is -14. The additive inverse of any number always adds up to zero when added together, which is why it is called the "additive inverse".
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What quadrant is 0degrees?
0 degrees is located in Quadrant I of the Coordinate Plane.
Quadrant I is the top right section of the coordinate plane, and it contains all the points where the x-coordinate and y-coordinate are positive. 0 degrees is the angle formed by the positive x-axis, and it is located in Quadrant I.
A coordinate plane is divided into four sections called quadrants, which are numbered I, II, III, and IV in a counterclockwise direction, starting with the top right.
This quadrant is defined by angles between 0 and 90 degrees and is generally associated with positive values on both the x- and y-axes.
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What is a 4 polygon?
A 4 polygon is a two-dimensional shape with four sides and four angles.
What is polygon?Polygon is a closed figure in a plane made up of straight lines or curved lines that intersect in pairs. A polygon has at least three sides and angles. The sides can be of any length and the angles can be of any measure. Polygons are classified according to the number of sides they have, such as triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), etc. Polygons are used in many fields, from mathematics to engineering.
It is a type of quadrilateral, which is a polygon with four sides. Examples of 4 polygons include squares, rectangles, rhombuses, and trapezoids. All 4 polygons have four sides of equal length, and all their interior angles add up to 360 degrees.
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Which of the following could be the units for dy/dx if y is the surface area of a tumor and x is the radius of the tumor
The units (derived unit) for dy/dx will be option C i.e. square millimeter per milimeter.
What are units?
Unit is a quantity of constant magnitude which is used to measure the magnitudes of other quantities of the same manner.
There seven SI base units, which are comprised of:
Length - meter (m)
Time - second (s)
Amount of substance - mole (mole)
Electric current - ampere (A)
Temperature - kelvin (K)
Luminous intensity - candela (cd)
Mass - kilogram (kg)
Derived units - A derived unit is a unit of measurement in the International System of Units (SI) that is derived from one or more of the seven base units.
As
Surface area(y) is mm^2
and radius(x) is mm
so, dy/dx will be mm^2/mm.
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ABCD is a parallelogram. Find the length of AB. A. 20 B. 17 C. 7
Change 20.4 yards to inches.
Answer: 734.4 inch
Step-by-step explanation:
Answer:
734.4 is the correct answer
how to graph y=-2/3x-5
-2x+3y=-15
By plotting the given points (0,-5) , (1,-17/3 ) for y=-2/3x-5 and the points for-2x+3y=-15 are (0,-5) , (1,-13/3) and (2,-11/3) we get the required graph.
What is linear equation ?
Linear equation can be defined as the equation in which highest degree is one.
Given Equation,
y=-2/3x-5 and -2x+3y=-15
So,
use the slope form is
y= mx+b , where m is slope and b is the y-intercept.
find the valued of m an b using the form y = mx+b
m = -2/3 and b=-5
The slope of line is the value of m and y-intercept the value of b
slope = -2/3
Any line can be graphed using two points. Select two x values, and plug them into the equation to find the corresponding 2y values.
create a table for x and y values.
x y
0 -5
1 -17/3
and soon
Similarly for -2x+3y=-15
x y
0 -5
1 -13/3
2 -11/3
and soon
And Hence plot the points in the graph.
Therefore, By plotting the given points (0,-5) , (1,-17/3 ) for y=-2/3x-5 and the points for-2x+3y=-15 are (0,-5) , (1,-13/3) and (2,-11/3) we get the required graph.
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Please help! Trig!!!!!!!l. Look at the image! And answer the question! Thank u!
Therefore , the solution to the given problem of trigonometry comes out to be Sin( θ) = Cos(90 - θ).
Describe trigonometry.Correlations between triangles' angles and sides are studied in the mathematics discipline known as trigonometry. Since every straight-sided form can be reduced to a collection of triangles, trig can be found throughout geometry. The relationships between trigonometric and other branches of mathematics, in particular calculus, either real or complex, integrals, square root, and integrals, are astonishingly complicated.
Here,
Below is a list of the trigonometric values that are missing:
Angle A = θ-90.
28° sin = 62° cos
Cos 33° equals Sin 57°.
Cos 31° equals Sin 59°.
Sin( θ) = Cos(90 - θ).
Therefore , the solution to the given problem of trigonometry comes out to be Sin( θ) = Cos(90 - θ).
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Write a function in terms of $t$ that represents the situation.
Sales of $10,000 increase by 65% each year.
A function in terms of t that represents the situation of sales that is of $10,000 and increase by 65% each year is t = 10,000(1.65)ⁿ.
What is function?An element of one non-empty set and an element of another non-empty set are said to have a special relationship when they are connected by a function in mathematics.
An assignment of an element from set Y to each element of set X, where set X is the domain and set Y is the codomain, is what is meant when a function is applied from a set X to a set Y.
We have given that f(t) is given by the Sales of $10,000 increase by 65% each year, i.e.
As sales of $10,000 increase by 65% each year.
The situation is represented by an exponential growth function.
The initial amount a = 10,000
The rate of growth r = 65% = 0.65
Therefore, the function will be:
t = a(1 + r)ⁿ
Where,
t represents the final amount.
a represents the initial amount
r represents the rate of growth
1+r represents the growth factor
n represents the time.
⇒ t = a(1 + r)ⁿ
⇒ t = 10,000(1 + 0.65)ⁿ
⇒ t = 10,000(1.65)ⁿ
Thus, a function in terms of t that represents the situation of sales that is of $10,000 and increase by 65% each year is t = 10,000(1.65)ⁿ.
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What is -2[6^2-(8-5)^3]
Answer:
Well yeah its -18
Step-by-step explanation:
Jacob is roller skating. He can skate 3/10 of a mile in 1/2 of a minute. What is Jacobs unit rate in miles per minute? please answer this is 100% of my grade TwT
Answer:
[tex]\frac{3}{5}[/tex] miles per minute
Step-by-step explanation:
3/10 ÷ 1/2
3/10 x 2 = 6/10 = 3/5
The Expression (t^6) (t^-14) is equivalent to t^n . What is the value of n
The equation (t6)(t-14) is identical to tn when n = 8. A mathematical expression is composed of numbers .
what is expressions ?You can multiply, divided, add, or take something away in mathematics. An expression is put together as follows: Expression, numeric value, and math operator A mathematical expression is composed of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) Contrasting phrases and emotions is possible. An algebraic expression is any mathematical statement that has elements, quantities, and an arithmetic operation. As an illustration, the phrase 4m + 5 consists of the terms 4m and 5, as well as the given equation's variable m, which are all split by the algebraic sign +.
given
(t^6) (t^-14) = t^n
= 6 -14 = n
n = 8
The equation (t6)(t-14) is identical to tn when n = 8.
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Does 15 12/9 make a triangle?
Answer: No it doesnt
Step-by-step explanation:
12+9 = 21 which is greater than 15 meaning it cannot create a triangle
You want to know how long the road marked x is and you know the lengths of the other two roads that form a right angle with the road, has the measurements to find the length marked x
What is the domain of f )= 4?
The domain of f(x) = 4 is the set of all real numbers.
The domain of any constant function (where the output is always the same) is the set of all real numbers. This is because any real number can be plugged into the function, and the output will always be 4.
The domain of f(x) = 4 is the set of all real numbers, including integers, rational numbers, irrational numbers, and any other types of real numbers.
The domain of f(x) = 4 is the set of all real numbers.
The domain of any constant function (where the output is always the same) is the set of all real numbers. This is because any real number can be plugged into the function, and the output will always be 4.
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Wanda sews small and large gloves. It takes her 45 minutes to sew a small pair of gloves and 120 minutes to sew a large pair of gloves. The costs of producing the gloves are $2 for a small pair and $4 for a large pair. Wanda has 16 hours available to sew gloves. The materials to make the gloves must cost at most $40. The system of linear inequalities represents this situation.
{45x+120y≤9602x+4y≤40
The point represents the number of small gloves is 16 and the number of large gloves is 2.
What is linear inequality?According to the mathematical definition of inequality, neither side is equal. When a relationship leads to a non-equal comparison between two expressions or numbers, it is known as an inequality in mathematics. The equal sign "=" in this statement can be replaced with any of the inequality symbols, such as greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤), or not equal to (≠).
Given Wanda sews small and large gloves,
time for small gloves = 45 minutes
time for large gloves = 120 minutes
total time = 16 hours = 960 minutes
let x be the number of small gloves and y be the number of large gloves,
inequality equation is 45x + 90y ≤ 960
and the cost of small gloves = $2
cost of large gloves = $4
total amount = $40
equation 2x + 4y ≤ 40
let us assume the system of inequality in form of a linear equation,
45x + 90y = 960
2x + 4y = 40
by solving the equations
the equation is written as
x = 20 - y
45(20 -y) + 90y = 960
y = 2
and x = 16
so the minimum values for x and y are 16 and 2 to maximize the profit Wanda should not decrease the production of small gloves from 16 and large gloves to 2.
Hence the point of maximization is (16, 2).
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the complete question is in image.
What is the value of x in the triangle? an isosceles right triangle with two legs of length 10 and hypotenuse of length x.
After using the Pythagorean theorem, the hypotenuse of the length x is 14.14 cm
In this case, we are given that:
The isosceles right triangle has two legs with length 10 cm
Let’s assume leg 1 as “a” = 10 cm, and leg 2 as “b” = 10 cm
The length of hypotenuse is “c”
To find the value of the hypotenuse or the longest side of a right triangle, we can use Pythagorean theorem:
a² + b² = c²
10² + 10² = c²
200 = c²
c = √200 = 14.14 cm
Therefore, the length of the hypotenuse would be 14.14 cm
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i need help hurryyy !!
Answer: 3
Step-by-step explanation: -4 1/3 divided by 2 1/5 = -1 32/33
Answer:
-1 32/33
Step-by-step explanation:
Negative divided by positive is negative.
We know the answeer will be negative.
Now ignore the signs, and just divide the mixed numerals.
1. Convert the mixed numerals to fractions.
2. Divide.
3. Convert quotient back to mixed numeral.
4. Make answer negative.
1. 4 1/3 ÷ 2 1/5 = 13/3 ÷ 11/5
2. = 13/3 × 5/11 = 65/33
3. = 33/33 + 32/33 = 1 32/33
4. Answer: -1 32/33
What 2 - 3
5319
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Can someone please help me
Therefore , the solution of the given problem of triangle comes out to be x = 32.376 side of the hypotunse length.
What precisely is a triangle?Since a triangle has four or even more sections, it is a polygon. It has an easy geometric form. Triangle ABC refers to a square with angles A, B, & C. A singular plane and square are produced by Euclidean geometry whenever the sides are still not collinear. A triangle is a polygon if it has 3 components and three corners. Where a triangle's three sides come together are known as the corners. A triangle's angles add up to 180 degrees.
Here,
Given:
Triangle having :
=> Sin 78° = h /30
=> 0.9781 = h /30
=> h = 30 * 0.9781
=>h =29.343
and
For x :
=> Sin 65° = 29.343 /x
=> 0.9063 = 29.343 /x
=> x = 29.343 / 0.9063
=> x = 32.376
Therefore , the solution of the given problem of triangle comes out to be
x = 32.376 side of the hypotunse length.
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Two sides of a triangle each measure 8 inches.
Maria says the triangle must be isosceles. Is Maria correct? Explain.
A.
No; the third side could measure 9 inches or 7 inches, and the triangle would be scalene.
B.
Yes; an isosceles triangle has at least two sides that have equal measures.
C.
Yes; the third side could measure 9 inches, and the triangle would be equilateral.
D.
No; the third side of the triangle could be 8 inches, and the triangle would be acute.
Answer:
B.) An isosceles triangle has at least two sides with equal measures.
Step-by-step explanation:
There are two equal sides and two equal angles in an isosceles triangle. There should be at least 2 equal sides for an Isosceles triangle. This means, even an equilateral triangle is an Isosceles triangle. In this case, at least two equal sides are 8 inches each. This is why the correct answer is 'Yes; an isosceles triangle has at least two sides that have equal measures.'
Lana want to encloe a garden plot each ide of the garden will be 9 feet what i the area of the garden
The area of the garden is 81 square feet (9 x 9).
Since each side of the garden plot is 9 feet, then the area of the garden plot can be calculated by multiplying 9 feet by 9 feet, which is equal to 81 square feet.
To calculate the area of the garden plot, we need to know the length and width of the plot. Since the length and width of the garden plot is both 9 feet, then the area of the garden plot can be calculated by multiplying the length and width together. 9 feet multiplied by 9 feet is equal to 81 square feet, which is the area of the garden plot.
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Which of the angles measures more than 90 but less than 180?
Answer:
Isosceles triangle
Step-by-step explanation:
Isosceles triangle has more than 90 but less than 180, which means Isosceles triangle is equal to 120.
The graph of a piecewise function is shown.
The graph shows a piecewise function in 3 pieces. The first piece is curved and increasing from the left through negative 3 comma negative 7 and ends at a closed point negative 2 comma negative 2. The second piece is linear with an open point at negative 2 comma negative 1 and another endpoint at 2 comma 1. The third piece is curved with an endpoint at 2 comma 1, and it goes through the point 3 comma 6 and continues to increase to the right.
What is the domain of the function?
(−∞,∞)
[−∞,∞]
(−∞,0)
(0,∞)
The domain of the piecewise function as described in the task content is; ( −∞ , ∞ ).
What is the domain of the described piecewise function?It follows from the task content that the domain of the piecewise function which is as described in the task content be determined.
The domain of a function is the set of all possible input values of that function.
For the first piece, since the curve increases from left ( -∞ ) through a point and ends at a closed point (-2, -2). Hence, the set of inputs for the first piece in interval notation is; ( -∞, -2 ].
Also, for the second piece which is linear with an open point at (-2, -1) and ends at an endpoint (2, 1). Hence, the domain of the second piece is; [-2, 2].
Finally, the third piece is curved with an endpoint at (2, 1) and continues to increase tonthe right. Hence, the domain of the third piece is; [2, ∞)
Ultimately, by combining all sets of input values as determined; ( -∞, -2 ], [-2, 2] and [2, ∞); the domain of the whole piecewise function is; (−∞,∞).
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How likely is the spinner to land a consonant rather than a vowel?
The likelihood of the spinner to land on a consonant rather than a vowel is C. Less likely.
How to find the likelihood ?The likelihood of the spinner landing on a consonant rather than a vowel can be found by the formula :
= Number of consonants / Number of outcomes on the spinner
= 4 / 9
= 0. 44
This likelihood is less than 50 % which is what would have made the likelihood equally likely. However, it is not much lower than 50 % so the likelihood would be less likely.
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find the sum of the measures of interior angles of the indicated convex polygon
(a) nonagon (b)dodecagon c) 23-gon ?
(a) The sum of the interior angles of the nonagon is 1260°.
(b) The sum of the interior angles of the dodecagon is 1800°.
(c) The sum of the interior angles of the 23-gon is 3780°.
What is a convex polygon?
A convex polygon is a closed figure with all of its interior angles less than 180° and all of its vertices pointing outward. The term convex refers to a shape with a curve or a projecting surface. To put it another way, all of the lines across the outline are straight and point outwards.
The formula for the sum of the interior angle is (n -2)180°.
The number of sides of the nonagon is 9.
Putting n = 9 in (n -2)180°, to find the sum of interior angle of nonagon:
The sum of the interior angle is (9 -2)180° = 1260°.
The number of sides of dodecagon is 12.
Putting n = 12 in (n -2)180°, to find the sum of interior angle of dodecagon:
The sum of the interior angle is (12 -2)180° = 1800°
Putting n = 23 in (n -2)180°, to find the sum of interior angle of 23-gon:
The sum of the interior angle is (23 -2)180° = 3780°.
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May I have help on this please? thanks if you do!
Answer: 1. linear equation
2. non-linear equation
3. non-linear equation
4. non-linear equation
5. linear equation
Step-by-step explanation:
not sure
I need help on quiz 4-1 classifying and solving for sides/ angles
Step-by-step explanation:
I don't know because you have it to far
simplify. (9x2- 8x-3) + (2x+8)
A. 11x + 10x +5
B. 9x2-6x+5
C. 9x2+10x-11
D. 11x2-6x-11
The result of simplifying the quadratic polynomial (9x2- 8x-3) + (2x+8) is 9x2 - 6x + 5.
what is polynomial ?In mathematics, the intersection point is the place on the y-axis where the slope of the line passes. a place where the y-axis crosses on a line or curve. Y = mx+c, where m stands for the slope and c for the y-intercept, is used as the equation for the straight line to show this. The slope (m) of the line and its y-intercept (b) are highlighted in the equation's intercept form. When an equation has the intercept form (y=mx+b), the slope is m and the y-intercept is b. Some equations can also be rewritten so that they appear to be slope intercepts. If y=x is rewritten as y=1x+0, for instance, the slope and y-intercept are both changed to 1.
given
(9x2- 8x-3) + (2x+8)
9x2 - 8x - 3 + 2x + 8
9x2 - 6x + 5
Thus, the result of simplifying (9x2- 8x-3) + (2x+8) is 9x2 - 6x + 5.
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