Given Ln 8 = = [(1 + (6 - 1) =) *co ( 7 + (1 - 1) ²)] Σ (i (i (i (i n a = b definite integral, that is provide a, b and f(x) in the expression in Sºf(² f(x)dx. b = express the limit as n→∞ as a f(x) =

Q: Suppose that f(x) is a function, whose derivative is f '(x)=(1 + cos x)(e* – 1)(x – 2)²(x – 3). At…

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Q: x²-1 1. Evaluate the derivative of f(x) = x -- approach XI at x = 0 using the limit

A: Note:-Since you asked multiple questions , according to our policy we can solve only first question…

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: (5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' =…

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Q: Use the product rule to differentiate the function h, where a and b are real numbers. h(x) = e¯ax…

A: According to product rule, if f(x)=u(x)*v*x) then f’(x)=u(x)*v’(x)+v(x)*u(x).

Q: Let g(x) be a function such that g(x) > 0 for all x, and g'(x) = 2g(x) for all x. A function f(x)…

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Q: Let F(x) –3 cos(ť²) dt. Use the Fundamental Theorem of Calculus to find F'(x). F'(z) =

A: Given: F(x)=∫x−1(−3cos(t2))dtF'(x)=ddx∫x−1(−3cos(t2))dt=?

Q: Assume that the function f defined by f(x) = 5x+ sin(Tx) is strictly increasing on R and find…

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Q: Suppose that f (0) = 2 and f'(x) 0. Show that f(4) 0.

A: f(0)=2   and f'(x)≤3  for all x>0 ⋯(i)integrate (i) w.r.to x ,we getf(x)≤3x+kput…

Q: == 3.14. Prove l'Hôpital's rule in the following form. Suppose f(a) = f'(a) = f(n-1) (a) = 0, g(a) =…

A: Given :f(a)=f'(a)=....=f(n-1)(a)=0  , g(a)=g'(a)=...=g(n-1)(a)=0 We have to prove :…

Q: Demonstrate that for the integral of p(x)/q(x) where p(x) is of degree 2 and q(x) of degree 3 it is…

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Q: 3. Compute the first four derivatives of f(x) = sin r + cos r. Try to find a pattern and guess the…

A: we have to find the first four derivative of f(x)=sinx + cosx and then we have to find the n-th…

Q: Show that r sin (1/x) fails to have a derivative at r = 0 and even the one-sided limits (lim,-0+ (f…

A: We will prove the given statement.

Q: VI. Suppose h' is continuous at x = 0 and h'(x) =tan(2x) - 1 if x > 0. Find h(0), if it exists.

A: Let's find.

Q: The limit represents the derivative of some function f at some number a. State such an f and a.…

A: Given: limh→0cos(π+h)+1h

Q: Let f(x) = (x" + 10x) e 6x-5, where n is an unspecified constant. Find the value of n such that the…

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Q: Q1. Derive the following formula for approximating the derivative f' and show that it is O(h) by…

A: Given: The formula is f'(x)≈112h -f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h). To prove: The formula and O(h4)…

Q: Define h(x) = x³ sin(1/x) for x + 0 and h(0) = 0. By using the product and chain rule we know that…

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Q: dy If y = e5z+14 then- dr O 5e5z+14 O 70e5z+14 O 4e5z+14 14e5 14 O 6e5z1 14

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Q: 6) When some variable z is a function of time (t), the derivative of z with respect to t is denoted…

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Q: 3. (2') Let A+ cos x x 2. Given that q(x) is a continuous function, determine the value of A and B.

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Q: -- ΣΧ E n² 22n n = 0 II

A: follow the next step

Q: If f(x) = (3x – 2)(2x + 3), then a. f(– 2) b. f'( – 2) = c. f''(– 2) d. The values of x for which…

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Q: Let f (x) =e*. sin(x²). what is the correct result of the derivative of f with respect to x?…

A: This question is related to differentiation, We will solve it using given information.