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1. (a) lim
x→π4 tan 2x · tan(π4 − x). (10points) (b) Suppose lim
x→∞f0(x) = A, a 6= 0, find limx
→∞{f(x + a) − f(x)}. (10points) 2. Suppose f (x) = x2 x ≤ 1
ax+ b x > 1 . Find a and b such that f is continuous and differentiable at x = 1. (10points)
3. Find the equation of the tangent line of the graph: y2 = x3+ 3x2 at (1, −2).
(5points)
4. Graph y = √x+1
|x−1|. Be sure to compute intervals of monotonicity, the inter- vals of concavity, the position of all local extrema, and inflection points, and all asymptotic lines. (15points)
5. Find dxdnn(√1+x1−x).(10points)
6. Given a sphere with radius r, find the height h of a pyramid of minimum volume whose base is a square and whose faces are all tengent to the sphere.
(10points) 7. (a)R x+1
√2x+1dx. (5points) (b)Rπ
0
psin3x− sin5xdx. (5points) 8. Prove or disprove
(a)If x3+ x = f (x) has at least two solutions, then there exists t such that f0(t) ≥ 1. (5points)
(b)We can find a non-constant differentiable function f defined for all x such that f0(x) = 0 for all n1, n=1,2,3,... (5points)
9. Let f (x) be a continuous function.
Define
F(x) = Z x
0
( Z u2
0
f(t)dt)du for x ≥ 0 G(x) =
Z x2 0
f(u)(x −√
u)du for x ≥ 0
Compute F0(x) and G0(x), and prove that F (x) = G(x) for x ≥ 0. (10points)
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