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Math 141 Homework # Due Tuesday, 10/2/ Extra Problems
These problems are taken from the Math 121 sample midterm exam from Fall 2005.
Problem #1 What value of x is f (x) = x^3 + 12 x^2 โ 2 x โ 3 decreasing most rapidly?
Problem #2 If c is a constant, then lim hโ 0
ech^ โ 1 h
equals
(a) ln c (b) c (c) ec^ (d) 0 (e) none of the above
Problem #3 Evaluate lim hโโ 4
h + 4 โ h + 6 โ
or explain why it does not exist.
Problem #4 Let f (x) =
x โ
x^2 โ 3
. Evaluate the following:
(#4a) lim xโ 1 f (x)
(#4b) lim xโ 3 f (x)
(#4c) lim xโ โ 3
f (x)
(#4d) lim xโโ 2
f (x)
Problem #5 Suppose that f and g are functions such that f is continuous, f (โ1) = 3, and lim xโโ 1
g(x) f (x)^2 + 1
- Find lim xโโ 1
g(x).
Problem #6 Suppose that f (3) = 2, f โฒ(3) = โ1, g(3) = 3, and gโฒ(3) = 5. Find the following numbers: (i) (f g)โฒ(3); (ii) (g/f )โฒ(3); (iii) the derivative of xโ^1 /f (x) at x = 3.
Problem #7 Let f (x) = x^2 + 1. Find every number a such that the line tangent to the graph of f (x) at the point (a, f (a)) passes through the point 91, 0).
Problem #8 The position of a particle at time t is given by s(t) = t^3 โ 4 t^2 + 3t for t โฅ 0.
(#8a) When is the velocity equal to 6? (#8b) When is the acceleration equal to 0? (#8c) When does the particle reverse its direction of motion?
Problem #9 Let f be the function defined by f (x) = 2x โ 1 for x โฅ 1 and f (x) = 3x โ 2 for x < 1. At a = 1, the function f is
(a) continuous (b) discontinuous because lim xโ 1 f (x) does not exist as a real number
(c) discontinuous because lim xโ 1 f (x) 6 = f (1)
(d) none of the above
Problem #10 Find an equation for the tangent line to the parametric curve (x, y) = (3 sin t, e^2 t) at the point (0, 1).
Problem #11 Find an equation for the tangent line to the curve 3(x^2 + y^2 )^2 = 14x^2 โ y^2 at the point (
Problem #12 Calculate d dx
[
x^2 + 3)sin^ x
]
Problem #13 The derivative of f (x) = cos(x^2 ) at x = 0 is given by the expression
(a) lim hโ 0
cos(h^2 ) โ cos h h
(b) lim hโ 0
cos(h^2 ) โ 1 h^2
(c) lim hโ 0
cos(h^2 ) โ 1 h
(d) lim hโ 0
cos h โ 1 h
(e) none
of the above
Problem #14 For which value(s) of c is the function f (x) defined below continuous everywhere?
f (x) =
c^2 x if x โค 1 c + 6x if x > 1
Problem #15 Suppose that the tangent line to the graph of f (x) at (โ 1 , 2) passes through the point (1, 5). Find f (โ1) and f โฒ(โ1).
