Calculus 11/14/2005
Name: ID:
1. For the limit lim
x→2
x
2 = 1, find the largest δ that ”works” for = 0.1. Ans:
2. Let f (x) = 2x − 1, x ≤ 2
x2− x, x > 2 . Find lim
x→2−
f (x) + f (2) + 3 lim
x→2+
f (x). Ans:
3. Find lim
x→4
√x − 2
x − 4 . Ans:
4. Let f (x) =
x2, x < 1
Ax − 3, x ≥ 1 . Find A given that f is continuout at 1. Ans:
5. Find lim
x→0
tan 3x
2x2+ 5x. Ans:
6. Solve the inequality 2x − 6
x2− 6x + 5 < 0 for x. Ans:
7. Find the rate of change of y = [x(x + 1)]−1 with respect to x at x = 2. Ans:
8. Find dy/dx at x = 2 if y = (s + 3)2, s =√
t − 3, t = x2. Ans:
9. If g(x) = f (x2+ 1), find g0(1) given that f0(2) = 3. Ans:
10. Find d2
dx2 x2sin 3x
Ans:
11. Find d dt
t2d
dt(t cos 3t)
Ans:
12. If x2+ y2= 4, use implicit differentiation to obtain dy
dx in term of x and y. Ans:
13. Find the equation of the tangent line to the curve x2+ xy + 2y2 = 28 at the point (−2, −3).
Ans:
14. Find d dx
√x2+ 1 x + 2
!
Ans:
15. A particle is moving along the parabola y2 = 4(x + 2). As it passes through the point (7, 6), its y–coordinate is increasing at the rate of 3 units per second. How fast is the x–coordinate changing at this instance? Ans:
16. Estimate f (5.4) given that f (5) = 1 and f0(x) =√3
x2+ 2. Ans:
