1- sin x a) Evaluate the limit by using L’Hopital's rule lim *x→7½ 1+ cos 2x b) Evaluate the integral by using the integration by parts rule [x sin xdx

Q: Ql) check the continuity of fX) at X=-1, 0,1.23 -1<x<1 x-4 2x-1, 3x-2 2, x=3 f(x) = 1sxs2 25x53 Q2)…

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Q: Let f(x)=8/x^2 on the interval [1,4], For 1<a<4, Construct a rectangle from x=1 to x=a, with height…

A: Consider the graph for f(x) and draw a rectangle with height f(a) at point x = a. From the fig. find…

Q: . Differentiate the function using product rule. f(x) = e*(e* + sin x + x²)

A: Given : fx = exex+sinx+x2

Q: For-1 <x < 1 let arcsin(x) f(z) = %3D Find the antiderivative F(x) of f(x) such that F(0) = 1.

A: We use substitution substitution method to find anti derivative

Q: a) Write down the definition of the derivative in terms of limits. Using this definition find (x² –…

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Q: Express the limit as an integral. lim (8(x)²-3(x) ³)Ax over [0, 3] /=1 X ) dx

A: Express the limit as an integral.limn→∞∑i=1n(8(xi∗)2−3(xi∗)3)Δx over [0, 3]

Q: Differentiate the function f. Use the form f'(x) = f(x) = 9x² + x + 8 f'(x) = = lim h→0 f(x +h)-f(x)…

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Q: Solve the Bernoulli equation: x2 dy/dx -2xy = 3y4

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Q: 1) Evaluate each integral by substitution: (In x) dx a)f(x+1)(x +2x)'dx b)fxVx² +9dx 「ズ dx 7x+5 dx…

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Q: (d) VT tanr - 2 V lim /4 VT x(4x – 7) Hint: divide both parts of the fraction by VT x and represent…

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Q: sin z Prove that lim Cosz - = 1 and lim = 0. Z→0 Z Z→0 Z Hint: use the definition of the derivative,…

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Q: Determine whether a given function is homogenous or not homogenous: 1. f(x,y) = sin x 2. f(x,y) = x…

A: Since there is a mistake in third question i do first two question as per our company guidelines…

Q: tan(1 – x) limx→1 2 2x +x – 3,

A: Limit

Q: Consider the weighted inner product defined by or functions f and g. (f,g) = [¹ f(x) g(x) dx, The…

A: To define the function inner_product(f, g), we can use the integral function to compute the…

Q: , for x 1 Let f(x) be a function de fined by f(x)3= Discuss the contuniuty of f andf".

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Q: Use logarithmic differentiation to find for y = x4* %3D dx 2x Use LaHopital's rule to calculate the…

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Q: lim x418 eª (sin (¹) + cos (¹)) (x - 2)²

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Q: sin x lim x→0* sin Vx X

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Q: Let/(x)=sin³ x+A sin² X

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Q: 111 Find y', y"y" and +6x" 4 -25⭑ y = x 5 + 2 y = 3 sinx 34 ③y= e + lux y of: 4 Find the limite by…

A: Question  3: 3)   Question 4:  Hope this helps!

Q: Q\ Use appropriate functions to guarantee that (). x sin if x # 0 h(x) = 0, if x = 0 is continuous…

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Q: In a sequence {1, 5, 15, 35, 70, 126}, what is the recursion formula for these numbers in this…

A: Given: a sequence 1, 5, 15, 35, 70, 126, ... To find: Recursion formula.

Q: Find the discontinuities, if any.  f(x) = sin (x^2 - 2)  f(x) = sec x lim    sin θ / θ2 θ→0+

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Q: Find the tangent equation for the function: f(x) = Vx2 + 5 at =2?

A: As per our guidelines we are supposed to answer  only one question. Kindly repost other question as…

Q: Let f'(x) = (12x41) sec²(3x4 - In(x)) 2x and suppose that f(1) = tan(3). Determine f(x).

A: Introduction: Integration is the process of identifying a function's antiderivatives, commonly…

Q: Theoretical assignment Consider hyperbolic tangent function

A: Given,                f(x)= tanh(x)=ex-e-xex+e-x,     x∈-2, 2and cut the interval -2, 2 into 4…

Q: If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

A: Proof:            Given: f' is a continuous…

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.