| 20. The Definition of e Illustrate the definition of the number e by graphing the curve y = (1 + 1/x)* and the line y = e on the same screen, using the viewing rectangle [0, 40] by [0, 4].

Q: The graph of y = f(x) has a y – intercept of 3 and the tangent line at the point (x, y) has an x-…

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Q: 34. Find the slope of the curve x^2 - 3xy y^2 - 4x + 2y + 1 = 0 at the point (1,-1 O 1/5 1/6 O 1/4 O…

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Q: 1) Find the legth of the curve; y = e from x =1 to x = 2.

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Q: he slope of the line tangent to the curve y = x -8x3 + 22x2 - 24x +9 at x=1 is (a) -2 (b) 2 (c) 0…

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Q: Which of the following describes the behavior of the graph of y = |x² – 16| at x = 4: a.…

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Q: (a) If g(x) = x2/3, show that g'(0) does not exist.(b) If a ≠ o, find g'(a).(c) Show that y = x2/3,…

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Q: the line 3x+ y=l is the tangent line for the graph of y= f(x) at x=1, then f'(1)= 1 (C) 1 (D) 3 A) 3…

A: The slope of the line y=mx+c is m. The slope is equal to the derivative of y i.e., f'y=m

Q: 33. Find an cquation for the line that is tangent to the curve y = (1-r)/(1 +r) at r = 2.

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Q: 2 Find the distance from the point B = [0 5 0 to the line 1 +t| 0 -3

A: Here we will find the distance between the given point and line,

Q: Consider a function h(u, v) = 4u? + v?. Find the level curves for k = n + 1,n + 2,n + 3, n + 4, n +…

A: A scalar function is a rule that is used to associate a number, or quantity, with a particular…

Q: For what values of a are the graphs of y = a* and y = 1 +x tangent at x = 0? Enter the exact answer.

A: Please see the explanation below

Q: Assume y = 3 · 5* is the equation for the blue line shown in the graph below, and that the black…

A: The equation of curve is:  y=3·5x

Q: .3 ax when x= For what values of a and b is the line -3x + y = b tangent to the curve y 4? b

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Q: 21. NP and PQ are tangent to circle M. NP = 2x-2x and PQ = x -2x + 16. Find the length of NP.…

A: Length of NP problem.....

Q: 6. Find the length of the curve y =-+ from x =1 to x=2. 9. 28x

A: The curve is given by the equation, y=x99+128x7  and the point is given from x=1 to x=2. To find :…

Q: 50 Find the length of the curve whose equation is given by r = e on 0 < 0 < 2T.

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Q: 0. Find the length of the curve y = x/2 + 8 on the interval (0, 2]

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Q: The tangent line to the graph of y 5x-x' when x 3 is y -22x +54 OB 8. y-12x +54 y=-9x+5 OD.

A: The equation of the curve is given by y = 5x-x3 To evaluate : The equation of the tangent line to…

Q: find the length of the curve on the given interval: y = 4/3 (x+1) ^ (3/2), [0,3]

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Q: d). Find the x coordinate of each point of inflection of the graph off on the open interval -3 <x <…

A: d) For a point of inflection of f, f"=0 This means the points will be the critical points of f'.…

Q: Q2/Find the length of the curve T = sino + cose between -s0s

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Q: y-e-x4cos4x ស

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Q: The line y=x+k is tangent to the graph of y = x²-2. Find the value of k. 17 (C) 16 33 15 (A) 16 31…

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Q: The slope of the curve f(x) =. - at x =1 is O a. 1 Ob. 13 3 -17 O d. 3 3. 1/3

A: Differentiate wrt x at given value

Q: 12. The slope of the curve y = x? -e* at its point of inflection is A. - In 2 B. 2- In 4 C. 2+ In 4…

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Q: The tangent line to the graph of y = g(x) at the point (4, 2) passes through the point (1, 8). Find…

A: topic - slope of a tangent line

Q: Given the curve h(x)=x2−7, find the point on h(x) is closest to the point (57,−7). Enter your answer…

A: topic -curve and distance from a point

Q: (a) If g(x) = x2/3, show that g'(0) does not exist.(b) If a ≠ 0, find g'(a).(c) Show that y = x2/3…

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Q: %3D 42. Find f(x) dx for each graph of y = f(x). (a) y. (b) y 2. 1. 3 aronh of flr) = 2x + 3 and

A: Given query is to find the valuebof intgy from the graph.

Q: 1. In the open interval (0, T1/2) , find the point where the slope of the curve is 4? y = tan x

A: Given the curve  y=tan x

Q: • Let S be the graph of f(r, y) = r² + y² – 2r + y for (r, y) E D = {(r, y) :2² + y² < 4 and z + y 2…

A: The function is fx, y=x2+y2-2x+y The region D is x, y: x2+y2≤4 and x+y≥0 The edges are x2+y2=4 and…

Q: 7. Find the length of the curve y= x'+ I from x = 1 to x . 12x Give your answer as a simplified…

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Q: Sketch a curve with the following properties. 0, for x> 12

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Q: w For what values of x is the tangent line of the graph of f(x) = 6x' + 27x² – 71x – 54 - parallel…

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Q: 2 For the graph of f(x) = ax + bx +4, the slope of the tangent at the point (3, 1) is 5. Determine…

A: Given that        f(x) = ax2+bx+4

Q: Q 8/ Find all points on the graph of f(x)= (x-3)(x-2) at which the -1 tangent line is horizontal.…

A: As per our guidelines we are suppose to solve only first question. Kindly repost the other question…

Q: (a) If g(x) =x2/3 , show that  g'(0) does not exist.(b) If a≠ 0 , find g'(a) .(c) Show that y= x2/3…

A: Given function is gx=x23. Differentiating the function with respect to x,…

Q: Find the exact length of the curve over the given interval. 4 3/2 X= 3 1sys4

A: Given curve is x=43y32, 1≤y≤4

Q: Which of the following is/are points on the curve y = x4- 8x3 + 22x2 - 24x + 9 where the tangent…

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Q: c) Find the exact lenght of the curve y = 100x – 100 between on the interval 0 < x < 11/2 Answer:

A: We can find the arc length as below.

Q: 2. Determine whether the following statements are true and give an explanation or counterexample.…

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Q: 3. Find the length of the curve y =- 6. 1 from x=2 to x=3 2x

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Q: 1. Any function y = f(x) can be expressed as x = f(y) in standard mathematical methods for solving…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: b The curve f(x) =ax + has inflection point at (2,27). a) -- Find the values of a and b.

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Q: 10. Find the points of inflection of the curve y f(x)=x³ -x.

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2. Consider the function y(x) = 5x² over the interval [0, 10]. (a) If the points A(0, y(0)), B(5,…

A: As per guidelines, we will solve first three subparts; Given function is yx=5x2  over the interval…

Q: fa is a positive integer, then the slope of the tangent to the graph of y = a²a - aln(ax + 1) at a =…

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Q: Find an equation of the tangent line to the graph of y = B(x) at x = 24 if B(24) = 0 and B'(24) = -…

A: Given,         y=Bx at x=24 if B24=0 and B'24=-14.

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.