State the Fundamental Theorem of Calculus (Part I and Part II). Hence or otherwise: (a) If u is a differentiable function of a over I, obtain a formula for d (Hint: You may use (c) By Using L'hopital's rule, Hence find (√1+ u^du) dt] . dz² (b) Evaluate i. Evaluate the limit lim 7-0 dz T lim #-3 I I fint dt = F(x) - F(3) for some antidervative F of sint) f(t). sin (22) + 7z² - 2z 2² (2+1)² sin t

Q: If lim (æ) – f(a) x - a | exists, then f is differentiable at a. Select one: True False

A:

Q: Ꮖ = 5.² for f(u) du. The function f(x) is defined on [0, ∞) and satisfies x³ = Find f(x). (Hint: Use…

A: Given query is to find f(x).

Q: 3. Show that f(x) = e describes a bijective function from R to (0, +∞0). (a) Show that f is a…

A:

Q: Consider the following proposed variation of Rolle's theorem. Let f: (a, b) → R be a differentiable…

A: We have defined a special real valued differential on (0,1) .

Q: The slope of a line passing through the point (a,b) not on f(x), and a tangent  to f(x) = x² can be…

A: Given curve is: f(x) = x2 The equation of the tangent line of the given curve is: y-y1=m(x-x1) where…

Q: 8. Define a) the derivative of f(x) with respect to r (in terms of a limit). b) the function f(z) is…

A: According to guidelines we solve only first three parts.

Q: du u (х, 0) — и(0,0) lim x→0 %| да (0,0)

A: Consider a function f(z)=u(x,y)+iv(x,y) such that     z=x+iy∆z=∆x+i∆y If the function is…

Q: calculate the limit of the function limT(æ → 0) [cosx/x2]]

A: To calculate the limit of the function.

Q: Use concepts learned in this module to complete this problem set: particle motion, applications…

A: Solve for the limit

Q: Give a pair of functions f: R →R and g : R →R such that g is not tangent to f at 0 and yet lim,0…

A: To give a pair of function f : ℝ→ℝ and g : ℝ→ℝ such that g is not tangent to f at 0 and yet limx→0…

Q: If f (x) is continuous e (a,b], we define | f (x) dx= lim | f(x) dx . on Xa+ A True B False

A: Solution

Q: In(x - 1)2 | 26. lim x→2 In(3 In(x - 1)

A: L=limx→2ln(x-1)^2ln(x-1)

Q: For each of the following sequences, if the divergence test applies, either state that lim an does…

A:

Q: lim f(k+h)+f(k) h→0 ds Q3) Using the definition calculate the derivatives h dk 1 S= k – k of the…

A:

Q: Given x2 if x 2 f(x) – f(2) a) Determine the left and right limit of the function b) For what…

A:

Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

Q: If the limit definition of a derivative can be used to find f′, thenwhat is the purpose of using…

A: The geometric meaning of the derivative  f'x=ddxfx is the slope of the line tangent to y at x…

Q: 6. ) Express the limit as a derivative and evaluate. x2021 - 1 lim x - 1

A: we have to solve the limit problem

Q: f(2) is continuos if lim f(z) =f(z,)

A:

Q: Show that if limn→∞ an = 0, then Θ(an + 1) = Θ(1).

A: