Calculus I Quiz 2 Dec. 9, 2009 Name: Student ID number:

Calculus I Quiz 2 Dec. 9, 2009

Name: Student ID number:

1. (30 pts; 2 pts for each problem; No partial credit) Find the general antideriva- tive of the given function (f (x) = F⁰(x)).

(a) f (x) = 1, F (x) = +C

(b) f (x) = x², F (x) = +C

(c) For x≥ 0, f(x) = √

x, F (x) = +C

(d) f (x) = x^1/3, F (x) = +C

(e) f (x) = e^x, F (x) = +C

(f) f (x) = e^4x, F (x) = +C

(g) f (x) = 2^x, F (x) = +C

(h) For x6= 0, f(x) = 1

x, F (x) = +C

(i) For x6= 0, f(x) = 1

2x, F (x) = +C

(j) f (x) = sin x, F (x) = +C

(k) f (x) = cos x, F (x) = +C

(l) f (x) = sec²x, F (x) = +C

(m) f (x) = sec x tan x, F (x) = +C

(n) f (x) = 1

1 + x², F (x) = +C

(o) For x6= 0, f(x) = x²+ 1

x , F (x) = +C

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2. (5 pts) Given that f⁰(x) = 1− 6x and f(0) = 8, find f(x).

3. (5 pts) Express the limit as a definite integral on the given interval.

nlim→∞

∑n i=1

2x_iln (1 + x²_i) ∆x, [2, 6], (x_i = 2 + i∆x, ∆x = (6− 2)/n)

4. (10 pts) Find the derivative of

•

∫ _x

0

√t sin t dt, x≥ 0,

•

∫ _x⁴

x²

√t sin t dt, x≥ 0.

5. (10 pts) Evaluate the definite integral

∫ 1 0

xe^x2dx

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6. (10 pts) Evaluate the definite integral

∫ ₂

−1

x− 2|x| dx

7. (10 pts) Evaluate

∫

ln x dx

8. (10 pts) Evaluate

∫

sin⁴x cos³x dx

9. (10 pts) Evaluate

∫ 1

√1 + x² dx. (Hint:

∫

sec θ dθ = ln| sec θ + tan θ| + C)

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