The L'Hopital rule and The derivative of the exponential function. lim (ex + x³)*:

Q: lim (1-cos x1/2) / ln(1+x) ? x-->0+

A: Here we use the series expansion of the functions cos (x) and ln(1+x):

Q: Find the derivative of f (x) = -6 tan (x). Enclose arguments of functions in parentheses. For…

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Q: 4. The natural exponential function is its own derivative.Explain what this means graphically. (Use…

A: We don’t know anything about derivatives that allows us to compute the derivatives of exponential…

Q: Use the Power Rule to compute the derivative. (Use symbolic notation and fractions where needed.) L-…

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Q: Use the Power Rule to compute the derivative. (Use symbolic notation and fractions where needed.) L-…

A: ddttn=n.tn-1

Q: sin(Tæ) lim x → 12 In(x – 11) -

A: The given limit,          limx→12sinπxlnx-11

Q: 1 Find the derivative of f(x) = - at x= 3 %3D (Type an integer or a simplified fractic 21 %3D

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Q: Compute the derivative ecos(x)arctan(x)

A: The given function ecosxarctanx.

Q: 4) Use logarithmic differentiation to find the derivative Of the given fünction.Show all steps.…

A: Consider the given function y=xx1+x24

Q: Ceka that passes through the Find and sketch an exponential equation of the form y = points (0, 8)…

A: Substitute the values of (x,y) in the function given and find out the equation.

Q: x-sinx 1) lim 2) lim-(x-). tan x 3): write an equation for the tangent line at x-3 of the curve: 1…

A: Answer

Q: Find an equation of the tangent line at x = 0. y = x5 + cos(x) .13 (Use symbolic notation and…

A: We have given equation  y=x13+cosx Put x=0  y=0+cos(0)=1 Point on the curve x1,y1=0,1…

Q: Discuss the relationship between the derivative dy/dx and (Lim) with drawing if it is required ?

A: See the attachment

Q: Find the derivative y=arctan(square root 1+x/1-x)

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Q: 5th derivative of y = log x^5 to the base 2

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Q: 75. F(x) has a negative derivative over (-, 2) and (5,9) and a positive derivative over (2, 5) and…

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: lim   ln(cosx) x → π/2

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Q: 4. Using the first derivative test to find the location of the local maxima of the following…

A: 4.The given function is, f(x)=x3-3x2-9x-1

Q: Q5/ * lim (x-3)(sin-'(x) – x) using L Hopital Rule

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Q: a) Find the derivative of the function using limits: f (x) = 2x+1 b) Find an equation of the tangent…

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Q: The tangent line at x = 5 to the graph of f(x) = x passes through the points (5, f(5)) and (6, 200).…

A: Given, f(x)=x³ Therefore, f(5)= 5³= 125

Q: log-(cot 1x) lim 1- cosh(2*)

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Q: 4. Using the first derivative test to find the location of the local maxima of the following…

A: Given function :

Q: l = lim x→0+ In(x)

A: No, its wrong. you are correct limx→∞ln(x)=-∞…

Q: feois Pdf find

A: f(x) is the probability density function if the area under the curve is one and f(x)>0

Q: An exponential function for which the derivative is al- ways negative.

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Q: Find the slope of the graph of the function at the given point. Use the derivative feature of a…

A: fx=sinxsinx+cosx Differentiating with respect to x, we get…

Q: Find the limit

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Q: Use the definition of the derivative to compute the derivative of the function f(x) = 1 – 6x² at an…

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Q: Find the function whose tangent has slope xln√x for each value of x>0 and whose graph passes through…

A: It is given that the slope of the tangent is equal to dydx. This question can be solved using the…

Q: 6 lim Sinx ex -1

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Q: , (a) Let f(x) = −x²-x. Use definition of the derivative Equation 3.4 to compute f'(x). (No other…

A: The objective of the question is determine the equation of tangent line to the given function.

Q: lim x→0 (sin3x sin5x/x2) *without use of deritivatives or l'hospital's rule*

A: See the details solution in below

Q: 13. Suppose a continuous function f and its derivative f' have values table. Given that f(1) 2, use…

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Q: 2 sin x lim In

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Q: Find the derivative of f(x) Give answer in exponential notation with positive exponents

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Q: cosh x 2. Find lim. ex

A: We have to find the limit

Q: log, (cot-x) 3. lim 1- cosh(2") X -00

A: limit.

Q: 4 Find the derivative of: – 9/x - Type your answer without fractional or negative exponents. Use…

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Q: 1 - cosh x lim x→0+ ln(x² + 1)

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Q: Use central-difference method to approximate the derivative of f(x) = In x² with value of x at 1.8,…

A: In this question we are given that f(x) =ln (x2) We have to find the value of f'(1.8) with the ∆x =…

Q: Find the derivative of f (x) = 5* In (x). Enclose arguments of functions, numerators, and…

A: Given; f(x)=5^x. ln(x)

Q: Define the funetion sle) - and (a) Find the function's fourth derivative at r

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Q: compute the derivative. 19. y=(2cosθ+5sinθ)9

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Q: 1. State the power rule. 2. For which values of n have we proved the power rule? 3. Based on your…

A: 1. The power rule of derivatives gives the following: For any real number n,      ddxxn=nxn-1

Q: Find delta value for the limit of x-->0 of tan5x/sinx when epsilon = 0.1

A: Introduction: The limit of a real-valued function is the base of differential calculus. A single…

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.