[SOLVED] Vv156 Honors Calculus II Assignment 4

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Exercise 4.1 [Ste10, p. 385]

• (1pt) If f is continuous on [a,b], use −|f(x)| ≤ f(x) ≤ |f(x)| to show that
• (1pt) Use the result of prevous part to show that
• Exercise 4.2 [Ste10, p. 395] The error function

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

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.

• (1pt) At what values of x does this function have local maximum values?
• (1pt) Find the coordinates of the first inflection point to the right of the origin.
• (1pt) Does this function have horizontal asymptotes?
• (1pt) 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 th

Exercise 4.9 [Ste10, p. 412] Evaluate the definite integral.