(a) Given the function f(x) = e"+2 – 2 i. Give the equation for the graph shifted 2 units to the left, 2 units downward and reflected about y-axis. ii. Show that the function f(x) obtained in (i) is one-to-one. Determine the inverse of the new function from (i). iii. State the domain and range of the inverse function obtained in (ii). (b) Using Intermediate Value Theorem, justify that the equation a4 – 2a² + } = 0 has two solutions in the interval [-1, 1].

Q: 1) Find the second derivative for the following functions: a) f(x) = 4x³ + 3x² + x – 14 b) f(x) =…

A: Solve derivative a fx=4x3+3x2+x-14b fx=4x-3+2x-4c fx=5ex-e3x

Q: Suppose that the differentiable function y = f(x) has an inverse, that the graph of f passes through…

A: first we find equation of line which passes through (2,5) and slope (m)=1/4 y-5=m(x-2)…

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A:

Q: 2 a) For the function f(x) = X-3 , find f'(1) by using the definition of derivative.

A:

Q: The finance department of Tayitu Company conducted a study of their weekly sales in the past years,…

A: NOTE: Refresh your page if you can't see any equations. . here we want to find the weekly sales at…

Q: For each of the given functions y = f(x). f(x) = 5, P(2, 1) (a) fınd the slope of the tangent line…

A:

Q: 2) The amount of money (in billions of U.S. dollars) that a country spends on research and…

A: Here the amount of money that a country spend on research and development is given as a function. We…

Q: Consider the following function. g(x) = -x2 + 9x + 6 Find the first and second derivatives of the…

A: The function is given by gx =-x2+9x+6 To evaluate : The interval where the function is concave…

Q: Find the derivative of the inverse function s(x) if f(x) = -x+ 1 1 r + 1

A:

Q: Find the equation of the line tangent to the graph of f at the indicated value of x. f(x) = 11 e* +…

A:

Q: In the function:  f(x)= (3x^2)ln(x) , x>0 What are the inflection points of the function?

A: To find inflection points of the function we need second derivative of the function.Use product rule…

Q: Suppose that the differentiable function y = ƒ(x) has an inverse and that the graph of ƒ passes…

A: Given,

Q: Find the first derivative of the function g(x) = V5 +x +4cos (2x).

A: The function f(x) is a rule that associates or fixes a number to the particular point x. If y=f(x),…

Q: (b) Sketch the graph of f (x) = ln(4 – x²). Use the first and second derivatives to identify where…

A: Consider the function  fx=ln4−x2

Q: Determine the equation of the line tangent to the graph of the following function at x = 0. g(x)…

A: The given function is

Q: Find the inflection points of the function f(x) = ln(x^2 + 1)

A:

Q: Let y = f(x) = x² +1, evaluate the derivative of the inverse function x = f-l(y) at y = 2.

A: Given a function, y=f(x)=x2+1 To find the derivative of the inverse function of f(x).

Q: 10. Find a function y = ax +bx+c whose graph has an x-intercept of 1, a y-intercept of -2, and a…

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

Q: x-2 For the function 10x) = ind f(X), the third denvative of t and i ), the fourth derivative ofT…

A: Please refer the attached image for complete solution.

Q: Q2: - Find the second derivative of the following function at x = 2 15 marks 1) y=+ V 2) y 3 (x +)…

A:

Q: Find the indicated derivative for the function. h"(x) for h(x) = 2x "5 – 6x -8 h"(x) =

A:

Q: Find the domain of function defined as follows. y = ln(x + 7)

A: Given       y = ln(x + 7)

Q: Let f be the function defined for x > 0, with f (e) = 2 and f', the first derivative of f , given by…

A: Given, f be a function defined for x>0 and the derivative of f is f'x=x2lnx Point e, 2

Q: Find the equation of the line tangent to the graph of f at the indicated value of x. f(x) = In x²;…

A:

Q: The location of the point of inflection in the graph of the function y = x ln x is

A: Given: y = x ln x

Q: 1. Consider the function f ( x ) = 1 + 1/10 x3 + 1/25 x5 .  Select any point ( a , f -1 ( a ) ) on…

A: Given function : f(x)=1+110x3+125x5 To solve the given problem first we need to find the inverse of…

Q: In certain conditions, the atmospheric pressure (p) in the height (h) is given by p=29e-0.000034h.…

A: It is given that the atmospheric pressure in the height is given by the function…

Q: 1. Sketch the graph of the function 4x f(x) = x E (-00, 00) 1+4x² Be sure to calculate the local…

A:

Q: Determine an equation for the line tangent to the graph of the given function at the indicated point…

A:

Q: Let f(x)= e^x^2 . Find an equation of the line tangent to the graph of f at the point the (1,e).

A: Tangent to the graph,

Q: 4)Find the inflection points of the function . f(x)=ln(x^(2)+1)

A: In this question we will find the point of inflection   we have given function   f(x)=ln(x2+1)…

Q: Solve for the derivative of the given functions using the Chain Rule formula. Let f(x) = √(x2 + 3x +…

A: (a) fx=x2+3x+1 Inner function : u=x2+3x+1. Outer function : y=u.

Q: 1 If f(6) = 3, and f'(6) 2 Let f'(x) be the inverse function of f(x). (f ')'(3) = Write the equation…

A:

Q: What is the first derivative with respect to x of the function g(x) = 3/9x O 1/2 9.

A:

Q: Find the derivative of the function m: f(x)= e^x^3+4

A: Topic = Derivative

Q: 1 4. Use the definition of the derivative to compute f'(x) given f(x) = 2x 3

A: Given,          The function is fx=12x-3 .

Q: 4. If y(T) = 1 – exp(-bNT), find the inverse function T = T(y). (Show me your work!)

A: Answer

Q: Find the third derivative f"" for the function f(x) = I-5

A: The first derivative of is given as: Also the derivative of a function in form of is: The given…

Q: Given the function y=x2−x. Find the derivative of the inverse function at x=1.

A:

Q: Let f(x) = 21 – x² The slope of the tangent line to the graph of f(x) || at the point (-4, 5) is The…

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Q: Find the first derivative f' and the original function f. f"(x) = x³ – 48x, f'(1) = 0, f(1) = 16

A: We have to find f'(x) and f(x).

Q: Find an equation of the line tangent to the graph of f(x) at the point (5,9): f(x) = e-3x-1o + 8

A:

Q: For each of the given functions y=f(x), find the slope of the tangent line to its inverse function…

A: Definition used -         Inverse of a function -          For finding the inverse of a given…

Q: Find the first and second derivatives of the function. 14. f(x)= x to the 7/2+x to the 5/3-x to the…

A:

Q: For the function y 8 f(æ) = 1+ x² ' a. Find the slope of the line tangent to its inverse function at…

A:

Q: Find the slope of the tangent line to its inverse function f^-1 at the indicated point P, and find…

A:

Q: Answer the following questions for the function k(x) = x³ + x + 1: (a) The derivative at x = - is…

A: The given function is kx=x3+x+1

Q: Find the derivative of the inverse function for y=x⋅3x provided x>0.

A:

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: