) In the following exercises, use direct substitution to show that each limit leads to the indeterminate form o/o. Then, evaluate the limit. (a). (b). lim x-3 x² - 2x-3 x²-9 √x+6-3 lim x-3 x-3

Q: Given f(x)=√x + 2 and g(x)= x³, find ()(x) and write the domain of (x) in in interval notation.

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Q: e Find the volume generated by revolving the area enclosed by the loop of the curve y² = x (x-3)²…

A: Solution is given below:

Q: Find the antiderivative F(x) that satisfies the given condition: f(x) = 4x² +5e"; F(0) = -2. Show…

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Q: The area of a rectangle is given by the function A(x) = 3x³ + 6x² + 4x + 8. If the length is defined…

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Q: Find the lengths of the functions of x over the given interval. Approximate if necessary. y=e^x on…

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Q: Determine the equation of the normal line to f(x) = x³3.5¯³x at x = -1.

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Q: 5. Prove the identity: cot(A + B) cot(A - B) = cot² B - tan² A tan² A cot² B-1

A: We have to prove the identity

Q: Evaluate d2y/dx2 at the point (0, -9). x = t3 - 3t y = 3t2 - 9

A: Given: x = t3 - 3t y = 3t2 - 9

Q: The pressure P (in kilopascals), volume V (in liters), and temperature T (in kelvins) of a mole of…

A: Here we will use the chain rule i.e., if u and v are function of t with f=uv then…

Q: Show that 16x4+9y12 lim (x,y)→(0,0) 4x²+3y6 does not exist.

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Q: For the following graph the equations and shade the area of the region between the curves. Determine…

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Q: Which of the following graphs represents (x-2)² + (y+4)² = 257 1. le - 4 - 7 . $ 4 2 2 2

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Q: Find the absolute maximum value and the absolute minimum value, if any, of the function.…

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Q: Explain how you would decide between using the substitution method or the integration by parts…

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Q: PRACTICE ANOTHER The graph shows the depth of water W in a reservoir over a one-year period as a…

A: Topic- Average rate of change of function over interval

Q: Set up an iterated triple integral equal to 3 3-x² x²+y² 3 2-√4-x²-y² using: Spherical coordinates…

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Q: Find the Fourier series for the following functions: f(t) = |sin t], -1 < t < n and The function is…

A: Solution is given below:

Q: The critical numbers r = -1 and x = 2 are found from a continuous derivative f(x). Given that the…

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Q: 4. S 10 dx 4x3-4x²+5x

A: Let's solve given integral by using quadratic factors.

Q: Approximate the area under the curve y = x from x= x³ with 6 subdivisions. = 2 to 5 using a Right…

A: The curve y=x3 from x=2 to x=5 we have to find the area under the curve using Right Endpoint…

Q: 5. Using f(x)=xln√x + 2x; Do each of the following: Find f(x) and f(x) a. b. Find critical values…

A: The given function is fx=x lnx+2x we have to find f'x and f''x use the derivative formulas,…

Q: Use mathematical induction to prove that the statement for all positive integers. + (2n-1) = n² a)…

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Q: A producer of pedal-powered ice cream makers has determined that the revenue from the production and…

A: Given: Rx = 46x- 0.04x2 , 0≤x≤900 To find the interval of increasing and decreasing, we have to find…

Q: Consider the region bounded by x=4-y, the y-axis and the x-axis. 1 2 3 4 The volume of the solid…

A: Given:-

Q: 5. Use the divergence theorem to evaluate the surface integral ff, F. ds, where S is the sphere x² +…

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Q: 1 3 2 + To find the blue shaded area above, we would calculate: b ľ f(x) dx = area Where: a = 2 3 4…

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Q: Consider the function f(x) = 9 x³ Let F(x) be the antiderivative of f(x) with F(1) = 0. Then F(x) =

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Q: If ¹f [ 2³ sin ædæ = −2³ cos x + 3x² sin æ+ 6x cos a - 6 [ f(x)dæ, then f(x) = a. x cos x b. sin x…

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Q: The graph shows the rational function f (x) and the logarithmic function g(x). f(x)…

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Q: Proor: Let † and g be one-to-one functions. a) Prove that (a) fog is one-to-one b) Prove that (b) (ƒ…

A: We will prove this using the concept of one to on

Q: The graph of y = sin x is just the graph of y = cos x shifted the right. unit (s) to

A: Topic:- trigonometry

Q: What is the 8th partial sum of 10,000-2,000+400-80+7 Round your answer to the thousandths place.…

A: The given sequence is: 10000-2000+400-80+... To find the 8th partial sum.

Q: 5 S³ƒ( en that wer= f(x)dx 10 3 is ['f f(r)dr? what is

A: Topic :- application of integration

Q: Define the force field F(x, y, z) = (xy, —-yz, xz). Find the divergence and curl of F. Let C be the…

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Q: Use the rules for sums of powers of integers to compute the sum. 22 [² j=8

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Q: Suppose that h(x) = f(x) g(x) and that we know the following f'(3) = -1, g(3) = 2, & g'(3) = -5.…

A: Topic:- calculus

Q: Find +62+7) dr 1(2²³/20 +C

A: Integration  ∫3x2+6x+7dx

Q: The graph of a function is given. Determine the average rate of change of the function between the…

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Q: Solve the following inequality: 2x + 4 8 x < 16 x < 10 x < 8

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Q: This exercise uses the radioactive decay model. The burial cloth of an Egyptian mummy is estimated…

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Q: Find an equation of the plane that contains the line x = −2 + 3t , y = 4 + 2t , z = 3 −t and is…

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Q: ₁ = 4-3i and Z₂ = 3 + 4i Use trigonometric forms to find Z₁Z2 and Z1 Z2

A: Topic:- application of complex numbers

Q: ical numbers r = -1 and x = 2 are found from a continuous derivative the second derivative is f"(x)…

A: Topic:- application of derivatives

Q: (i) Consider the function f: R\{-2,3} → R defined by 2x³4x²5x - 3 x²-x-6 f(x)= - (a) Prove, using…

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Q: On Integration of Rational Fractions. Distinct linear factors Repeated linear factors x+8 7) 2+ 1²²…

A: 6) Given:       x+8x2+x-2=x+8x-1x+2 The partial fraction decomposition is:…

Q: 2 s x² dx x2-4

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Q: Find the volume of the solid generated by revolving about the y axis the region under the curve…

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Q: 25. The vector field F(x, y, z) = (2xy + tan x)i + (x²-√y² - 1)j is conservative. a) True b) False

A: To determine whether the following statement is True or False.The vector field F→x, y, z=2xy+tan…

Q: Consider the region R bounded by the functions f(x)=3x and g(x)=x²+2. The area of R is: Choices: sq.…

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Q: 8 √₁ 2 4-x² 4 x² 2 +4.x² dx

A: