Try It Yourself#
Exercise 1#
Compute
Verify your answer by computing its derivative.
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Answer: \(2ey + C\)
Exercise 2#
Compute
Verify your answer by computing its derivative.
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Answer: \(\frac{4}{3}x^{9/4} + 14\sqrt{x} - \frac{1}{24x^4} + C\)
Exercise 3#
Compute
Verify your answer by computing its derivative.
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Answer: \(-\frac{1}{2}e^{-2x} - \frac{20}{3}e^{3x} - \frac{4}{5}e^{-5x} - 80x + C\)
Exercise 4#
Compute
\(\int \frac{10x^3 - 2x^2 + 3x-25 }{x} ~dx.\)$
Verify your answer by computing its derivative.
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Answer: \(\frac{10}{3}x^3 - x^2 + 3x - 25\ln|x| + C\)
Exercise 5#
Find \(f(x)\) such that
Verify your answer by computing its derivative.
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Answer: \(\frac{1}{7}e^{7x} + \frac{1}{2}x^4 + 3x + \frac{13}{7}\)
Exercise 6#
The marginal cost function associated with producing \(x\) croissants is given by
where \(C'(x)\) is measured in dollars/unit and \(x\) denotes the number of croissants.
If the daily fixed costs incurred in the production is \(\$400\), find the total cost \(C(x)\) incurred in producing the first 100 units of the day.
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Answer: $3900
Exercise 7#
Compute
Verify your answer by computing its derivative.
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Answer: \(-\frac{1}{6(5x^2 - 6x + 15)^3} + C\)
Exercise 8#
Compute
Verify your answer by computing its derivative.
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Answer: \(x + 2\ln|x-2| + C\)
Exercise 9#
Compute
Verify your answer by computing its derivative.
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Answer: \(\frac{1}{3}\ln|e^{3x}-4| + C\)
Exercise 10#
Find \(f(x)\) such that
Verify your answer by computing its derivative.
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Answer: \(\frac{1}{6}(5x^2+2x-1)^3 - 21\)
Exercise 11#
Compute
Verify your answer by computing its derivative.
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Answer: \((x^2+10)^{3/2} + C\)
Exercise 12#
Compute
Verify your answer by computing its derivative.
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Answer: \(3(x^2+10)^{5/2} - 50(x^2+10)^{3/2} + C\)