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(a) Consider the integral exp(t) dt. Use the substitution arcsin(t) = u to formulate this as an integral in terms of u. (Do not attempt to evaluate the integral.)

(a) Consider the integral exp(t) dt. Use the substitution arcsin(t) = u to formulate this as an integral in terms of u. (Do not attempt to evaluate the integral.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 30EQ
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(a) Consider the integral exp(t) dt. Use the substitution arcsin(t) = u to formulate this as an integral in terms of u. (Do not attempt to evaluate the integral.) (b) Suppose that q(x) is periodic with period p (so that q(x) = q(x +p)). Use integration by parts (with g(x) = q(x)) to show that d (x)q(x) dr = 0. (c) Gallium-67 is a radioisotope that is used in medical imaging to detect tumours. Normally, when injected into the body, gallium will collect in various places such as the bones and liver. Cancer cells take up gallium more easily than normal healthy tissue, so a gallium scan can reveal the sites of possible cancerous tissue. It is therefore important to know the levels of gallium that should be in the body after a certain time. The half life of gallium-67 is 78 hours. Recall that the levels of Gallium-67 (in ppm) will satisfy G'(t) = =kG(t) and therefore be given by G(t) = Goe-kt. (i) Determine the value of k. (ii) What is the rate of change of the level of gallium when the level itself is 1000 ppm?
Transcribed Image Text:(a) Consider the integral exp(t) dt. Use the substitution arcsin(t) = u to formulate this as an integral in terms of u. (Do not attempt to evaluate the integral.) (b) Suppose that q(x) is periodic with period p (so that q(x) = q(x +p)). Use integration by parts (with g(x) = q(x)) to show that d (x)q(x) dr = 0. (c) Gallium-67 is a radioisotope that is used in medical imaging to detect tumours. Normally, when injected into the body, gallium will collect in various places such as the bones and liver. Cancer cells take up gallium more easily than normal healthy tissue, so a gallium scan can reveal the sites of possible cancerous tissue. It is therefore important to know the levels of gallium that should be in the body after a certain time. The half life of gallium-67 is 78 hours. Recall that the levels of Gallium-67 (in ppm) will satisfy G'(t) = =kG(t) and therefore be given by G(t) = Goe-kt. (i) Determine the value of k. (ii) What is the rate of change of the level of gallium when the level itself is 1000 ppm?
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