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For some practice working with complex numbers: Calculate $$(3 - 2i) + (4 + 4i) = \square$$ $$(3 - 2i) - (4 + 4i) = \square$$ $$(3 - 2i)(4 + 4i) = \square$$ The complex conjugate of $$(1 + i)$$ is $$(1 - i)$$. In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point in the complex plane by reflecting through the $x$-axis. The complex conjugate of a complex number $z$ is written with a bar over it: $$\bar{z}$$ and read as "$z$ bar". Notice that if $$z = a + ib$$, then $$(z)(\bar{z}) = |z|^2 = a^2 + b^2$$ which is also the square of the distance of the point $z$ from the origin. (Plot $z$ as a point in the "complex" plane in order to see this.) If $$z = 3 - 2i$$ then $$\bar{z} = \square$$ and $$|z| = \square$$ You can use this to simplify complex fractions. Multiply the numerator and denominator by the complex conjugate of the denominator to make the denominator real. $$\frac{3 - 2i}{4 + 4i} = \square + \square i$$ Two convenient functions to know about pick out the real and imaginary parts of a complex number. $$Re(a + ib) = a$$ (the real part (coordinate) of the complex number), and $$Im(a + ib) = b$$ (the imaginary part (coordinate) of the complex number. $$Re$$ and $$Im$$ are linear functions - now that you know about linear behavior you may start noticing it often.

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3 - Teenagers (13-19) LEARNING OBJECTIVE: Determine the nutritional ne < O O O > How many grams of fiber per day would you recomm a.) 15 b.) 25 c.) 10 d.) 18 SUBMIT MY ANSWER

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Question 8 of 10 In females, estrogen and progesterone: Select all that apply. stimulate gastrulation in the developing embryo. maintain adult health. direct development of female secondary sexual characteristics during puberty. direct development of female reproductive structures during embryogenesis. regulate the menstrual cycle and support pregnancy. Incorrect Answer

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Evaluate the integral by interpreting it in terms of areas. 7 52 x − 15 dx0

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Which of the following will not shift the supply curve for labor to the right? Multiple Choice A decrease in marginal income tax rates An increase in the wage rate An increase in the dollar cost of leisure A decrease in government unemployment benefits

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Which of the following is a type of autosomal recessive genetic disorder? A. Haemophilia B. Skeletal dysplasia C. Sickle cell anaemia D. Down syndrome

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The blood stream senses elevated blood calcium. You must complete the following steps to proceed: Identify the RECEPTOR responsible for monitoring this STIMULUS Correctly identify the CONTROL CENTER to receive the sensation input Identify the INSTRUCTIONS to be sent to the EFFECTOR Instructions should be hormonal related. This could have multiple steps if multiple effectors are involved. Identify the response of the EFFECTOR organ Choose the correct type of feedback loop and identify the end goal of this process.

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In the scientific method, a hypothesis has more data to support it than a theory. True False Submit Request Answer

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2. A 5 mL container with a hole having a radius of $5 \times 10^{-6}$ m is filled with hydrogen. This container is placed in an evacuated chamber at 0 $^\circ$C. How long will it take for 90% of the hydrogen to effuse out. (Hint: Remember flux () is equal to where A is the area of the hole, N is number of molecules, V is total volume and t is time)

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The problem of making choices regarding what to produce and in what quantities can be illustrated with a possibilities curve. Goods that bring us immediate satisfaction, like food, mag- azines, and CDs are called \textit{consumer} goods. Goods like machines, that give little immediate pleasure but that help increase output of goods and services in the future, are called ______ goods. When a production possibilities curve is bowed outward from the origin it is because of the law of ______ cost.

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