MATH 296 PROBLEMS 2

IGOR KRIZ

1. Solve:

x + 2y + z + t  + 3u = 10
z + 10t − 2u = 5
t  + 9u = 2

2. Let f : [a, b] → ℝ be continuous and let f′(t), f′′(t),...,f^{(n+1)}(t) exist for all t ∈ (a, b). Put

F (t) = f(b) − f(t) − \frac{f′(t)}{1!}(b − t) − \frac{f′′(t)}{2!}(b − t)^{2} − ... − \frac{f^{(n)}(t)}{n!}(b − t)^{n}.

(a) Show that for t ∈ (a, b),

F′(t) = −\frac{f^{(n+1)}(t)}{n!}(b − t)^{n}.

(b) Apply Lagrange’s theorem

\frac{F (a) − F (b)}{g(a) − g(b)} = \frac{F′(t)}{g′(t)}, t ∈ (a, b)

to the function g′(t) = (b − t)^{n+1}. Using (a) and F (b) = 0, conclude that

f(b) = f(a) + \frac{f′(a)}{1!}(b − a) + ... + \frac{f^{(n)}(a)}{n!}(b − a)^{n} + \frac{f^{(n+1)}(t)}{(n + 1)!}(b − a)^{n+1}

for some t ∈ (a, b) (Taylor’s expansion formula).

3. Find a polynomial f of degree ≤ 3 such that f(−1) = 4, f(0) = 3, f(1) = 4, f(2) = 19. Can you find all polynomials of degree ≤ 4 satisfying these equalities? [Using the equalities, formulate a system of linear equations for the coefficients of the polynomial.]

4. A garden is to be designed in the shape of a circular sector (see picture), with radius R and central angle θ. The garden is to have a fixed area A. For what value of R and θ (in radians) will the length around the perimeter be minimized?