Calculus 11/14/2005
Name: ID:
1. Which of the following pairs of functions are inverse functions of each other on the implied domains? (may have more than one answer)
A) f (x) =|x|; g(x) = |x| B) f (x) = x1; g(x) = x1, C) f (x) = 1x; g(x) = 1x, D) f (x) =√
x; g(x) = x2, for x≥ 0.
2. Which of the following curves is NOT the graph of a function?(may have more than one answer)
(A) (B) (C) (D)
A) graph A, B) graph B, C) graph C D) graph D
3. Find lim
x→1
√x− 1 x− 1 . 4. Let f (x) =
{ x2, x < 1
Ax− 2, x ≥ 1 . Find A given that f is continuout at 1.
5. Find lim
x→0
tan 3x 2x2+ 5x.
6. Find all discontinuities of f (x). For each discontinuity that is removable, define a new function that removes the discontinuity.
f (x) =
{ sin x
x if x6= 0 2 if x = 0
7. Find the rate of change of y = 1/[x(x + 1)] with respect to x at x = 2.
Ans:: −365
8. Find dy/dx at x = 2 if y = (s + 3)2, s =√
t− 3, t = x2. Ans:: 16. Hint:dydx = dydsdsdtdxdt
9. If g(x) = f (x2+ 1), find g0(1) given that f0(2) = 3.
Ans:: 6. Hint: g0(x) = f0(x2+ 1)· (2x) 10. 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).
1
11. Using the definition of detivative (limits), compute f0(x).
f (x) =√ x + 2
12. Find d2 dx2
(
x2sin 6x)
13. If x2 + y2 = 4, use implicit differentiation to obtain dy
dx in term of x and y.
14. Find the equation of the tangent line to the curve x2 + xy + 2y2 = 28 at the point (−2, −3).
15. Find d dx
(√ x2+ 1 x + 2
)
, d
dx(3ex2), d
dx(xx2), d
dx(sin−1x2)
16. 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?
17. Find the absolute maximum and absolute minimum values of the function f (x) = x3x+2
on the interval [0, 2].
18. Given f (x) = x2x−1, find:
• Domain of the function;
• Horizontal and Vertical Asymptotes;
• Interval of increasing and decreasing;
• Critical points and local extrema;
• Determine where the graph is concave up and concave down and locate any inflection points;
• Locate x- and y- intercepts, if any;
and draw a graph of the function showing all significant features.
• Double-Angle
sin 2θ = 2 sin θ cos θ cos 2θ = 2 cos2θ− 1 = 1 − 2 sin2θ
• Derivative formulas
d
dxsin x = cos x, dxd cos x =− sin x,
d
dxsin−1x = √11−x2, for −1 < x < 1 dxd cos−1x =−√11−x2,for−1 < x < 1
d
dxtan−1x = 1+x1 2, dxd cot−1x =−1−x1 2,
d
dxsec−1x = |x|√1x2−1,for |x| > 1 dxd csc−1x =−|x|√1x2−1for |x| > 1
d
dxex = ex
2
