4. Let a be a real number such that 0 < a < 1, and let f(t) = eat – t where t e R. a) Find the equation of the line which is tangent to the graph of f at the point (0, f(0) b) Does f have an inverse function? Explain your answer with some supporting evidence.

Q: t is d

A: It is Given that function is differentiable at x=3. and f'(3)=-6 and f(3)=1

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A: Given function is : fx=x2-1x-2 To verify the Rolle's Theorem and find the point in the interval…

Q: By considering the derivative of the function f: [-1,1] → R defined by f(r) = show r2 +1' that f-1…

A:

Q: 10. a) If f (x) = 3x² – 2x + 4 , find the Difference Quotient (z+h)-f (z) %3D in simplified form. b)…

A: Given the function f(x)=3x2−2x+4

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A:

Q: Find the derivative of the following function g2 F(x) = ds a 3+ 5s4 using the appropriate form of…

A: Given that,

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A:

Q: 5. By considering the derivative of the function f : [-1,-1] → R defined by f(x) = show that f-¹…

A: Concept: inverse function: An inverse function, sometimes known as an anti function, is a function…

Q: 1. Let f(x) = mx + b, the line with slope m and y-intercept b. Use the rules of differentiation to…

A:

Q: Suppose the linear approximation for a function f(x) at a = 2 is given by the tangent line y = -3x +…

A:

Q: 2. Find the derivative and the analytic domain of f(2) = 1+ %3D 22+1

A:

Q: Find the function f given that the slope of the tangent line at any point (x, f(x)) is f'(x) and…

A: Find the antiderivative of the given function and then the values of x and y to obtain the C. The…

Q: Consider a quadratic function f(x) = ax?+ bx + c, where a, b, CER and a # 0. She hat the tangent…

A: f(x)=ax2+bx+climit definition of derivative states that :f'(x) =limh→0f(x+h)-f(x)hf'(x) =limh→0…

Q: Let f(x) be a function that is differentiable at x = -2. Given that f'(-2) = 3 and f(-2) = 6, find…

A: Given data: f'(-2) = 3 x1 = -2 f(-2) = y1 y1 = 6

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: f(a) = ca + 4 if a E (-∞, 2] I ca -4 if a e (2, 00) if a E (2, ∞)

A:

Q: [8] Use the definition f(a +h) – f(a) f'(a) = lim h-0 h To find the derivative of the given function…

A: Given: So,          f'(2)=limh→0f(2+h)-f(2)h       =limh→ 012+h-12h       =limh→ 0-12h+2       Plug…

Q: 1. Decide if there is an analytic function f (x + iy) whose real part satisfies Re(f) = xe".|

A: Given problem: To decide analytic function fx+iy whose real part satisfies Ref=xey.

Q: Let y = 5−2x be the tangent line to a function y = f(x) at x = 3 and y = 2+3x is the tangent line to…

A:

Q: Find the absolute extrema of the function on the closed interval.

A:

Q: 2. Consider the function f(x) = 1 – x², z ER. Draw the graph of f and calculate the equation of line…

A: solve the following

Q: 2.4 Use the definition of the derivative of complex function to show that f(z) is not dif-…

A: The Complex Derivative Definition: Let f be a function defined on a neighborhood of z0 ∈ ℂ. If…

Q: Assume that the functions f and g are differentiable with f(2)=4, f'(2)=3, g(2)=-1, and g'(2)=2.…

A: Given,           Assume that the functions f and g are differentiable with f2=4,  f'2=3,…

Q: C. Is the function define by g(x) = x² – sin x + 5 continuous at x = n?

A:

Q: Let f(x) be defined by f(x) = 3x² + 5x + 2. Find the slope of the tangent line to the curve y = f(x)…

A:

Q: Let f(x) = 33 –x The slope of the tangent line to the graph of f (=) at the point (-5, 8) is The…

A:

Q: Suppose the linear approximation for a function f(x) at a = 3 is given by the tangent line y -2x +…

A: The given functions are

Q: 3. Let f be defined by f(x) = e* sin x. What is the slope of the line tangent to the graph of f at x…

A:

Q: Find the values of k in f(x)=x^2-kx such that the line y=5x-4 is tangent to the function.

A:

Q: 1 Let f(x)=- (3x – 5)² -6 and f'(x) = %3D (3x – 5) Find an equation of the line tangent to the graph…

A: The given function is: f(x)=1(3x-5)2

Q: 3. Find the lincar approximation of the function g(x) = V1+x at a = 0 and use it to approximate the…

A: Use taylor form

Q: At what other point does the tangent line drawn to the curve f (x) = x ^ 3 from (2,8) intersect?

A: Given curve is : f(x)=x3   ..............(1)           and tangent drawn from (2,8)

Q: Let f(x) = 9 – x² The slope of the tangent line to the graph of f(x) at the point (-3,0) is The…

A:

Q: d) Hence form a conjecture for the derivative of functions in the form y = (ax + b)². e) Prove your…

A: As you have mentioned I am answering Question (d) and Question (e)

Q: 5) Use the tangent line at x = 9 to approximate the value of the function f(x)= Vx at x= 8.8.

A:

Q: 2. Using the definition of derivative, prove the following: 1 (a) f^(x)= if f(x) = Vx %3D

A: fx=x3

Q: 1. For the function f(x) = 3x? + 8x – 2, (a) Find the slope of f(x) at x = 3 | (b) Find equation of…

A: According to the bartleby guidelines, for more than one questions asked, only the 1st question…

Q: 11. Use the definition of the derivative to find f'(x) for the function f(x) = x2 + x + 2. Show your…

A: We have to find the derivative of the given function.

Q: A) Consider the function f(x) whose second derivative is f’’(x)=4x+3sin(x). If f(0)=4 and f’(0)=4,…

A: According to the guidelines in case of multiple questions, We are only allowed to do first question.…

Q: Show the function f(z) = 1+2i+2Re(z) is differentiable or not differentiable at any points.

A: Solution

Q: The tangent line to y=f(x) at (7,0) passes through the point (6,−7). Compute the following. a.)…

A: Topic:- calculus

Q: 1. Verify whether the function f(2) = e=Y sin x – ie- cos x is entire or not. 2. Show that the…

A: Given: Let us consider the given, 1) fz=e-ysinx-ie-ycosx 2) fz=xy+iy

Q: 5. Use the definition of the derivative to show that the given function is no where differentiable.…

A:

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

A:

Q: 5.6 Differentiate f with respect to t and g with respect to X, where

A: Let us start solving step by step....differentiate with respect to t and g with respect to x

Q: 2. Consider the function below and the point (x, y, z) shaded in red. Is fa positive, negative, or…

A:

Q: 2 Consider the following piecewise function: f(x) = 0 4 x²+3' Use the first principles definition…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: (6) _If f'(a)=0, then the tangent of the function f at x=a is a horizontal line.

A: Since you have asked multiple questions in single request  so we  will be answering only first…