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Calculus Techniques

by: Benton Yundt

Calculus Techniques MATH 1100

Benton Yundt
Utah State University
GPA 3.66

Joseph Koebbe

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Joseph Koebbe
Class Notes
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This 8 page Class Notes was uploaded by Benton Yundt on Wednesday October 28, 2015. The Class Notes belongs to MATH 1100 at Utah State University taught by Joseph Koebbe in Fall. Since its upload, it has received 7 views. For similar materials see /class/230415/math-1100-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15
Problem De nition Problem 39 Present Value A business is expected to yield a continuous ow of pro t at the rate of 500000 per year If money will earn interest at the nominal rate of 9 per year compounded continuously what is the present value of the business for 20 years and b forever Present Value is de ned in Section 62 of the text Solution Step 1 The present value calculation for part is Present Value That s a lot of money Solution Step 2 3920 f 500 000glt0 09 dt 0 3920 500000 alt 009gt dt 0 v EH 09 20 900000 0 09 30 000 mew 0 09x20 Eg o 090 For part b we are trying to nd the present value of the company for The model is to take If out to in nity the foreseeable future Present Value 3500 000glt0 09 dt 395 500000 lim get00W dt b mc 0 6 0 09 b 009 I Eg o 09 Eg o 090 500 000 lim b mc 323 This is due to the fact that the exponential approaches zero the the argu ment tends to 00 Problem De nition Problem 33 Evaluate the inde nite integral 1 Solution Step 1 We want to substitute to remove complicated expressions However there is an easier That is divide the t in the denominator into the terms to form 2 t 1 262 1 LUV dt 7 dt t t Qt V2 t 1dt t12 m 4t 2 2m C 4v 2m C 2 ln t C Solution Step 1 The solution can be Checked by differentiating the inde nite integral to make sure we get back to the original integrand iv W C 4rt ln t C i 12 i i 4 t dtlntdtC 1 12 1 42t t0 1 1 12 2f f 2t1212t1 t 39 t Problem De nition Problem 71 Chemistry Carbon Dating The remnants of an acient re in a cave in Africa showed a carbon14 decay rate of 31 counts per minute per gram of carbon Assuming that the decay rate of carbon14 in freshly cut wood corrected for changes in carbon14 in the atmosphere is 136 counts per minute per gram of carbon calculate the age of the remnants The halflife of carbon is 5715 years Use the integrated rstorder law lnNNg kt where N is the number of nuclides present at time t N0 is the number of nuclides present at time t 0 k 06935715 and t is the time of the re Solution Step 1 For the present problem we can write 0693 1 1 7t mas gtlt 36 5715 This becomes 14787 00001212t or t m 12194 years Problem De nition Problem 21 Use the Log Rule to nd the inde nite integral 2 17 33 1 dr Solution Step 1 The problem requires that we perform a bit of algebra to get the integral in the correct form We start with u 33 1 This is the entire expression in the denominator This requires that we have du 7 2 dr 3933 The integral can be written 13 r31 amp quot r31 amp l dudr dr 11 C 7 u applying the generalized log rule for computing antiderivatives Solution Step 2 The next step is to return to the original variables The result is 1 1 gmmC m LC Solution Step 3 We can easily check the result by differentiating the antiderivative 13 d ma a 1 C d i ln r31 13 C WM WH WiH Problem De nition Problem 45 Pro t A sporting goods manufacturer produces regulation soccer balls at two plants The costs of producing an units at location 1 and 32 units at location 2 are given by Own 002 431 500 and respectively If the product sells for 45 per unit then the pro t function for the product is given by P3T13T2 45CC1l39 32 01x1 02C Find a P250 150 and b P300300 Solution Step 1 The evalauation for P250 150 is P250 150 45250 150 01250 02150 To nish this off we need 01250 0022502 4250 500 2750 and 02150 0051502 4150 275 2000 So P250 150 45250 150 2750 2000 45250 150 2750 2000 13250 The pro t for this setup is 13250 Solution Step 2 The evalauation for P 300 200 is P300 200 45300 200 01300 02200 To nish this off we need 01300 0026002 4300 500 3500 and 02200 01352002 4200 275 3075 SO P800 200 45500 3500 3075 15925 The pro t for this setup is 15925


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