Suppose the derivative of ƒ is ƒ'(x) = x - 3.a. Find the critical points of ƒ.b. On what intervals is ƒ increasing and on what intervals is ƒ decreasing?

Q: If ƒ(x) = x^n, n>=1, show from the definition of the derivative that ƒ′(0) = 0.

A: Given,

Q: Find the equation of the tangent line to the graph of y at (2, 금). Use the limit process in finding…

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Q: Find the derivative of the following functions using the long method (limits) (a) x =t-5t- 3 (b) y =…

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Q: Determine the intervals along the x-axis on which the derivative in Figure 16 is positive. y 4.0 3.5…

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Q: Use backward, forward, and centered difference approximation to estimate the second derivative of…

A: We will do all three parts.

Q: find the derivative of f (x) at the designated value of x. f (x) = 1x at x = 1/16

A: Given: fx=1x To find the derivative  Differentiate the given function with respect to x we get,…

Q: Use the first derivative test to find the location of all local extrema in the interval for the…

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Q: Evaluate the derivative at x = 1 for y = In ()

A: For a function f its derivative is defined as f'(x)=limh→0f(x+h)-f(x)h. For some standard functions…

Q: find the derivative of y with respect to the appropriatevariable. y = (1 - θ) tanh-1 θ

A: Consider the following equation:

Q: find the derivative of y with respect to the appropriatevariable. y = √(s2 - 1) - sec-1 s

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Q: Suppose the derivative of the function y = f(x) is y' = (x- 2) (x + 6). At what points, if any, does…

A: see 2nd step

Q: Suppose that the SECOND DERIVATIVE of a function f is given by f" (x) = (x – 2)² (x² – 4x). Find the…

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Q: Determine the derivative of f(x) = ( In x )6 with respect to x. f'(x) =

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Q: find the derivative of y with respect to the givenindependent variable. y = xπ

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Q: find the derivative of f (x) at the designated value of x. f (x) = x3 at x = 12

A: The derivative of a function can be calculated by differentiating both sides of the function with…

Q: Find the general anti-derivative of f(x) = + Vx on a continuous interval.

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Q: find the derivative of y with respect to the appropriatevariable. y = sinh-1 √x

A: Find the derivative of y with respect to the appropriate variable.

Q: Develop a general rule for the nth derivative of xf(x), where f is a differentiable function of x.

A: Given: the function xfx, where f is a differentiable function of x.

Q: Find the derivative of the function by the limit process. h(s) = −2 (square root of x)

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Q: Use forward and backward difference formula with h = 0.1 to approximate the derivative of f(x) =…

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Q: Let ƒ(x) = √2x + 1. Compute ƒ'(2),the derivative of ƒ at x = 2, and use the result to find an…

A: Consider the given function, fx=2x+1 differentiate with respect to x, f'x=ddx2x+1=122x+1·2=12x+1…

Q: p(x) = In(x)^/x

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Q: Match the graph of the above function with the graph of its derivative. 0 ㅇ O 수 +

A: The given graph of the function is the graph of a semi circle. Or positive side of the circle.

Q: find the derivative of f (x) at the designated value of x. f (x) = x + 11 at x = 0

A: To find the derivative of f(x)=x+11 at x=0 Solution: f(x)=x+11 Differentiating both sides w.r.t x we…

Q: 2. Determine the true error and relative true error when approximating the derivative of f (x) = -x²…

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Q: Suppose that the second derivative of the function y = f(x) is y" = x*(x = 2)°(x + 3). For what…

A: Given: The second derivative of the function y=f(x) is y"=x2(x-2)3(x+3) Definition: The inflection…

Q: Suppose that the first derivative of y = ƒ(x) is y' = 6(x + 1)(x - 2)2. At what points, if any, does…

A: Given the first derivative is: y'=6x+1x-22

Q: Give a justified example of function f(x) that is differentiable everywhere, yet its derivative has…

A: Definition of Differentiability p(x) is said to be differentiable at the point x = b if the…

Q: find the derivative of f (x) at the designated value of x. f (x) = x5 at x = 3/2

A: Given function is: fx=x5 We have to find the derivative of f(x) at x=32. Then we get,…

Q: 3. The derivative of f(x) = In x at x = 3 when h = 0.002

A: 3) To find the true error and relative error of the function: f(x)=ln x, x=3 & h=0.002

Q: Suppose that the first derivative of y = ƒ(x) is y'= 6x(x + 1)(x - 2). At what points, if any, does…

A: The derivative of the function y=f(x) is y'=6xx+1x-2. We have to find at what points, if any, does…

Q: show that the function is derivable and that the derivative is continuous.

A: The objective is to show that the given function is derivable and the derivative is continuous.…

Q: Shows that if a function f(x) is continuous in interval [a,b] then most likely the derivative of…

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Q: What role does the denominator of f'(x) play in finding the critical points (in other words, what do…

A: The given data is: f'x

Q: 2. Prove from the definition of derivative that for n EZ, d ng"-1 dr

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Q: Let y be a function of x and suppose y= In(y-x). What is the correct result for the derivative of y…

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Q: Suppose the derivative of f is f '(x)=- At what values of x does the f does have a x slope of -4.

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Q: find the derivative of y with respect to x, t, or θ, as appropriate.

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Q: f(x) = x*Inx

A: TO FIND 5th order derivative f(x) = x4 lnx FORMULA USED  BIPART PRODUCT RULE OF DIFFERENTIATION…

Q: Find the first and second derivatives of the function. f(x) = x/(7+x2) Identify the intervals on…

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Q: Suppose the derivative of the function y=f(x) is y’=(x—1)^2(x—2). At what points, if any, does the…

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Q: find a linearization at a suitably chosen integer neara at which the given function and its…

A: Consider the following function: fx=x2+2x Differentiate fx with respect to x: ddxfx=ddxx2+2xf'x=2x+2

Q: Suppose that the first derivative of y = ƒ(x) is y′ = 6(x + 1)(x - 2)2. At what points, if any, does…

A: To determine if the function has any points of maxima, minima or inflection, use second derivative…

Q: 3 Find the derivative with gicapeet to X If f(0 = 0-7 The derivatie with reapeet to X If -7X is…

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Q: Suppose that the first derivative of y = ƒ(x) is y′ = 6x(x + 1)(x - 2). At what points, if any, does…

A: Here the first derivative of the given function is y’= 6 x (x+1) (x-2). If a function has zero…

Q: Find the derivative off with respect to x of f(x) = In x⁹. The derivative off with respect to x of…

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Q: Use the definition of derivative to compute f '(0). e-1/x if x * 0 if x - 0 f(x) f '(0) -

A: The definition of the derivative is given by: f'(x)=limh→0f(x+h)-f(x)h⇒f'(0)=limh→0f(h)-f(0)h  ...…

Q: find a linearization at a suitably chosen integer neara at which the given function and its…

A: Given: fx=xx+1 We want to find a linearization at a suitably chosen integer near a at which the…

Q: suppose the derivative of f exists, and assume that f(2)=3, f'(2)=4. Let g(x)=x^2f(x) and h(x)=…

A: The objective is to find the equation of tangent lines of g(x) and h(x) at x=2

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.