1. Differentiate. Show your work to get credit. a) g(x) = x* In(2.x² +7) by using the product rule.

Q: Find the limit of the square root of (4x^(6)-x) over (x^(3)+3), as x approaches positive infinity

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Q: -25 Evaluate the integral: dax x²Vx2 +25

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Q: 4.3 Use the calculus of residues to evaluate the following integral (x2- dr. + 1)2

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Q: 4. Integrate the following function using the Trapezoid Rule with n=5: (1.2

A: As indicated in the question, we will make use of composite Trapezoidal rule to approximate the…

Q: Use the Simpson's Rule to evaluate the integral fx|x²dx 1

A: Given integral is ∫−11xx2dx. We have to evaluate the given integral by using Simpson rule. Here,…

Q: Why are special methods, such as l’Hôpital’s Rule, needed to evaluate indeterminate forms (as…

A: l’Hôpital’s Rule are needed to evaluate indeterminate forms

Q: The area bounded by the curve y 7x and the lines x = 9 and x = 42 above the x х — ахis - is_ O…

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Q: The limit as x approaches infinty of (x+3^x)/(x^2-3^x) Evaulate using L'Hospital's Rule providing a…

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Q: Find the length of the graph of f(x) = cosh(x) for 0<x< In(12).

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Q: Is it possible for φ(n) (the Euler's totient function) to be bigger than n?

A: The objective is to find the possibility (if any) for φ(n) (the Euler's totient function) to be…

Q: (b) Verify Mean Value Theorem for f(x)= 3x² – 7x – 4 over the interval[–1,4] .

A: Given: fx=3x2-7x-4

Q: 4. Why is f'(x) approximated well by the difference quotient when Ax is very small?

A: Derivative of function f by definition is: f'x=lim∆x→0fx+∆x-fx∆x This is the difference quotient…

Q: 4. Verify Rolle's Theorem for f(x) = x² – 2x +1 on [-1,3]

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Q: 2. Will the limit of all polynomial functions / (x) as x approaches a be always equal to the value…

A: Find your answer below

Q: 1. Use the limit definition of the derivative to find g'(x) if g(x) = 2x² – 6x + 13.

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Q: 66. What will be the function g(x) resulting from normalizing the function f(x) = x³ − x4 defined in…

A: To find : function gx resulting from normalizing the function fx=x3-x4 defined in the domain 0, 34

Q: 4. Verify the Mean Value Theorem for the function y=ln (2x + 3) in

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Q: What is the x=g(x) that will diverge the root of f(x) = x3 + 4x2 - 10?

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Q: 21. Discuss the continuity of: for for 0sx 2

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Q: B)Is w = Zez analytical function or not?

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Q: Еxercise 13 Determine how тany times 0 арpears in thе тultiplicatioп table for Z16-

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Q: Evaluate the limit as x approaches infinity of (x2 - ln(3/x)) / 4x2+6x

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Q: 5. Apply L'Hopital's rule to determine the limits of the function. y = lim e"(x – a² + 9)

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Q: 2. Estimate V50 using tools from calculus.

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Q: Show that the following continuous function has a root between l and 2. f(x) = 4x - 6x² + 3x - 2

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Q: Txーt e lim 4x² 4,2 +4x

A: Given that limx→0 e2x-14x2+4x We have to find the limit using L-Hopital rule L-Hopital's rule: If…

Q: Show that L₁J - dx x² + 2x + 2 = In(√5-2).

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Q: Show that x + is increasing on [1, 0).

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Q: Prove that, cosh-'x = In |x + vx² – 1 for x 2 1 -기/

A: This question is based on Cosine hyperbola.

Q: Verify the Mean value Theorem for the function f(x) = x² – 6x + 5 on [0, 6]

A: mean value theorem

Q: Estimate f(3.2) by using the following Data obtain from the f(x) = sinx by use Quadratic…

A: Given function fx=sinx. The objective is to estimate f3.2 by using quadratic interpolation.

Q: - use the Cauchy Residue Theorem to integrate the given function around the

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Q: (b) Verify mean value theorem for f(x)=x² - 5x+6 over the interval [1,3].

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Q: What is the domain of g(x)=ln(25x-x^2) Show all the work.

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Q: Compare the graphs of the functions. Y1 = In(2x) and Y2 = In 2 + In x

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Q: Find the extrema of the function g(x) = x3 using any of the methods given in the above discussion.

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Q: Proved that cosh fas/ (cashZ+L)

A: To prove :  cosh2z2 = 12cosh (z) + 1

Q: Verify mean value theorem for ƒ(x)=x' – 5x+6 over the interval [1,3].

A: Please refer to the image below

Q: Suppose G, (x) and G2(x) are both antiderivatives of a function g(x). Using complete 3. sentences,…

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Q: 27 Use the Fundamental Theorem of Calculus to evaluate S (Vx2 -) dx. 8.

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Q: 8(x) = X -x3 + 2x x4

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Q: Verify the Mean Value Theorem for f(x) = In(6x + 1) on | 0, %3D

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Q: The term r ryis homogenous of degree two. -

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Q: (i) Prove that Lim Vent +1 1 1 + + + /2n²+n] ... V2n2+2

A: Given that, g(x)=limn→∞12n2+1+12n2+2+....+12n2+n from sandwich theoram we can say if f(x)≤g(x)≤h(x),…

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.