# Is the equilibrium value y a b similarly show that

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) is the equilibrium value y = a / b . Similarly, show that the average value of the population x ( t ) is the equilibrium value x = c / d . 2. Evaluate the integrals 5 3 + x dx and 5 3 + x 2 dx by hand. Also use your CAS to evaluate 5 3 + x 3 dx , 5 3 + x 4 dx and 5 3 + x 5 dx . Describe any patterns you see. In particular, are there any constants with recognizable patterns? What types of functions appear? If the arguments of the logarithms are multiplied, what is the result? Conjecture as much as possible about the form of 5 3 + x n dx for positive integer n . 3. Physicists define something called the Dirac delta δ ( x ), for which a defining property is that b a δ ( x ) dx = 1 for any a , b > 0. Assuming that δ ( x ) acts like a continuous func- tion (this is a significant issue!), use this property to evaluate (a) 1 0 δ ( x 2) dx , (b) 1 0 δ (2 x 1) dx and (c) 1 1 δ (2 x ) dx . Assuming that it applies, use the Fundamental Theorem of Calculus to prove that δ ( x ) = 0 for all x = 0 and to prove that δ ( x ) is unbounded in [ 1, 1]. What do you find trouble- some about this? Do you think that δ ( x ) is really a continuous function, or even a function at all? 4. Suppose that f is a continuous function such that for all x , f (2 x ) = 3 f ( x ) and f ( x + 1 2 ) = 1 3 + f ( x ). Compute 1 0 f ( x ) dx . 4.7 NUMERICAL INTEGRATION Thus far, our development of the integral has paralleled our development of the derivative. In both cases, we began with a limit definition that was difficult to use for calculation and then, proceeded to develop simplified rules for calculation. At this point, you should be able to find the derivative of nearly any function you can write down. You might expect that with a few more rules you will be able to do the same for integrals. Unfortunately, this is not the case. There are many functions for which no elementary antiderivative is available. (By elementary antiderivative, we mean an antiderivative expressible in terms of the elementary functions with which you are familiar: the algebraic, trigonometric, exponential and logarithmic functions.) For instance, 2 0 cos( x 2 ) dx