Exercise 4.1 [Ste10, p. 385]
(a) If f is continuous on [a,b], use −|f(x)| ≤ f(x) ≤ |f(x)| to show that
(b) Use the result of prevous part to show that
Exercise 4.2 [Ste10, p. 395] The error function
erf
is used in probability, statistics, and engineering. (a) (1pt) Show that
x2erf(x) satisfies the differential equation y′ = 2xy + 2/√π. (b) (1pt) Show that the function y = e
Exercise 4.3 [Ste10, p. 396] The sine integral function
Si
is important in electrical engineering. [The integrand f(t) = (sint)/t is not defined when t = 0, but we know that its limit is 1 when t → 0. So we define f(0) = 1 and this makes f a continuous function everywhere.] (a) (1pt) Sketch the graph of Si.
(b) At what values of x does this function have local maximum values?
(c) Find the coordinates of the first inflection point to the right of the origin.
(d)Does this function have horizontal asymptotes?
(e) Solve the following equation (for x) correct to one decimal place:
Exercise 4.4 [Ste10, p. 396] Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0,1].
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Exercise 4.5 [Ste10, p. 396] If f is continuous and g and h are differentiable functions, find a formula for
d Z h(x)
f(t)dt dx g(x)
Exercise 4.6 [Ste10, p. 396] Find a function f and a number a such that
for all x > 0.
Exercise 4.7 [Ste10, p. 414] Evaluate the indefinite and definite integral.
ZZ √
tanx 22
(i) e sec xdx(iii) cotxcsc xdx
Z π/3
(vi) x4 sinxdx(viii)
−π/3
Exercise 4.8 [Ste10, p. 412] If f ∈ C0(R), show that
Exercise 4.9 [Ste10, p. 412] Evaluate the definite integral.
