Please help me with this. Please check if I've gotten the write answer for each questions. And if I have shown the correct steps. Just check question d) and e)Just write the answers that are correct (e.g. a) correct, b) incorrect )Image 1: The questionImage 2: My work i XZ4Differentiate with respect to 'x'f'(x) = (x² + 3) dx (√x) - √x dx (x² + 3)(x²+3) ²UXd) f(x) = x²+31X-74+2√x (x² + 3) = 2x √x(x² + 3)²Therefore2x² + 3-4x²2√x (x²+3) ²-3-3x²Therefore when x 74f'(x) = 3-3x2√x (x²+3) ²(+),e) ((4) = (4) ² +3 = 1613 = 2lim f(x)X-74limX-742= 2√x (x²+3) ²24+15limX-74+fin =++16+3(x+15)219X²+3219lim f(x) lim f(x) = f (4)X-74X-74+(using Quotient Rule)13121Thus the function f(x) is continuous at x = 4 Consider the following piecewise function: f(x) =XCOLE x<00≤x≤4a. Use the first principles definition of a derivative to determine f'(x) when 0 < x < 4b. Determine f"(3)c. Determine the equation of the tangent to f(x) when x = - =d. Determine f'(x) when x > 4e. Is f(x) continuous when x = 4? Use the formal definition of continuity to argue your answer...

Answered: Image /qna-images/answer/6fd0b6b7-eaf6-4ff9-94f1-10a4a5394d0e.jpg

Consider the following piecewise function: f(x) = Xcos, x<0 0sx<4 a. Use the first principles definition of a derivative to determine f'(x) when 0≤x<4 b. Determine /"(3) c. Determine the equation of the tangent to f(x) when x = - d. Determine f'(x) when x>4 e. Is f(x) continuous when x = 4? Use the formal definition of continuity to argue your answer.

Consider the following piecewise function: f(x) = Xcos, x<0 0sx<4 a. Use the first principles definition of a derivative to determine f'(x) when 0≤x<4 b. Determine /"(3) c. Determine the equation of the tangent to f(x) when x = - d. Determine f'(x) when x>4 e. Is f(x) continuous when x = 4? Use the formal definition of continuity to argue your answer.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Please help me with this. Please check if I've gotten the write answer for each questions. And if I have shown the correct steps. Just check question d) and e)

Just write the answers that are correct (e.g. a) correct, b) incorrect )

Image 1: The question

Image 2: My work

i XZ4
Differentiate with respect to 'x'
f'(x) = (x² + 3) dx (√x) - √x dx (x² + 3)
(x²+3) ²
UX
d) f(x) = x²+3
1
X-74+
2√x (x² + 3) = 2x √x
(x² + 3)²
Therefore
2
x² + 3-4x²
2√x (x²+3) ²
-3-3x²
Therefore when x 74
f'(x) = 3-3x
2√x (x²+3) ²
(+),
e) ((4) = (4) ² +3 = 1613 = 2
lim f(x)
X-74
lim
X-74
2
= 2√x (x²+3) ²
2
4+15
lim
X-74+
fin =
++
16+3
(x+15)
2
19
X²+3
2
19
lim f(x) lim f(x) = f (4)
X-74
X-74+
(using Quotient Rule)
13121
Thus the function f(x) is continuous at x = 4

Consider the following piecewise function: f(x) =
XCOLE x<0
0≤x≤4
a. Use the first principles definition of a derivative to determine f'(x) when 0 < x < 4
b. Determine f"(3)
c. Determine the equation of the tangent to f(x) when x = - =
d. Determine f'(x) when x > 4
e. Is f(x) continuous when x = 4? Use the formal definition of continuity to argue your answer...

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