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STAT/MATH 511 HOMEWORK 0
- Compute the following limits:
(a) lim x→ 2 x^2 − 4 x − 2 (b) lim x→ 0 x sin x (c) (^) xlim→∞ x^2 e−x.
- Find all values of x satisfying f (x) = 0:
(a) f (x) = x^2 + 2x + 2 (b) f (x) = 2x+2^ − 3 x−^1 (c) f (x) = ln(4x − 7).
- Compute the following integrals:
(a)
∫ (^1) 0
x^2 (1 − x)dx (b)
∫ (^10) 0
e−x/^2 dx (c)
∫ (^) e 1
x
dx.
- Find the derivatives of the following functions:
(a) f (x) = eax^2 +bx+c (b) g(u) = u ln u − u (c) h(y) = ln(y^2 − 4 y).
- Compute the following derivatives:
(a) d^3 dx^3 xex^ (b) d^2 dt^2 et+2t^2.
- Find the following sums:
(a)
∑^ ∞ n=
)n (b)
∑^ ∞ j=
)j (b)
∑^ ∞ j=
)j− 1 (d)
∑^9 y=
)y− 1 .
- Evaluate the following double integrals:
(a)
∫ (^1) 0
∫ (^) x 0 2 xy^2 dydx (b)
∫ (^1) u
∫ (^1) u/y 2 1 dy 1 dy 2 , 0 < u < 1.
- Compute the following integrals:
(a)
∫ (^) ∞ 0 xe−x^ dx (b)
∫ (^) ∞ 0 x^2 e−x^ dx (c)
∫ (^5) 0 x^2 e−x/^2 dx.
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STAT/MATH 511 HOMEWORK 0
- Evaluate the following:
(a) lim x→ 0 xex^ (b) (^) xlim→∞ xex^ (c) lim x→ 0 xe−x^ (d) (^) xlim→∞ xe−x.
- Suppose that f (x) = 4x^2 − 1. Write a formula for f −^1 , the inverse of f. Graph f and f −^1 on the same set of axes. Do the same with g(x) = e−x^ and h(x) = log 3 (x − 1).
- Compute the following limits:
(a) (^) nlim→∞
( 1 +
n
)n (b) (^) nlim→∞
( 1 −
n
)n .
- Discuss the increasing/decreasing behavior of the functions (a) f (x) = 1 − e−x/^2 and (b) f (x) = x^2 e−x. Restrict attention to x > 0. Also, discuss concavity.
- Compute the following sums:
(a)
∑^ ∞ j=
2 j j! (b)
∑^ ∞ j=
j (c)
∑^ ∞ j=
( |x| 1 + |x|
)j .
- Compute the following integrals:
(a)
∫ (^) ∞ 0
∫ (^) x+ x
e−y^ dydx (b)
∫ (^1) 0
∫ (^1) 0
y 1 e−(y^1 +y^2 )^ dy 2 dy 1.
- Find the derivatives of the following functions and evaluate their derivatives at t = 0:
(a) f (t) = (0.3 + 0. 7 et)^10 (b) f (t) = e^10 t+t (^2) / 2 .
- Compute the following integrals:
(a)
∫ (^1) 0 ex
1 − ex^ dx (b)
∫ (^) ∞ 0 y^3 e−y (^4) / 2 dy.
- Find the Taylor Series expansion of f (x) = sin x about x = 0.
- Write (x + y)^6 in its binomial expansion.
- Find both partial derivatives of f (x, y) = x ln(xy). Also, compute the Hessian of f (x, y); i.e., the matrix of second partial derivatives.
- Show that
∑^ ∞ x=
(x + 1)(x + 2) = 1 and
∑^ ∞ x=
x (x + 1)(x + 2)
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