Calculus
Fall Final Review I
Name:_________________________________
f ( x) =
1. Find all points of inflection:
1 4
x − 2 x 2 + 15 .
12
2. A farmer has 160 ft. of fencing to enclose 2 adjacent rectangular pig pens. What dimensions should be
used so that the enclosed area will be a maximum?
3. The radius of a sphere is measured to be 3.0 inches. If the measurement is correct to within 0.01 inch,
use differentials to estimate the propagated error in the volume of the sphere.
4. Find all critical numbers:
=
f ( x) x 2 x + 1
y 8 x 3 − 2 x 4 is concave downward.
5. Find all intervals for which the graph of the function=
) 2 x + 5cos x on [0, 2π].
6. Locate the absolute extrema: g ( x=
7. Find the critical numbers and the open intervals on which the function is increasing or decreasing:
y=
− x3 + 3x 2 − 2
x + ∆x − x
∆x
x
9. Find lim
x → 0 sin 3 x
8. Find: ∆lim
x →0
−12t + 51t + 38 . Which statement is true?
10. The position function for a particular object is s (t ) =
a. The velocity at t = 1 is 27
b. The velocity is constant
c. The initial position is 51
d. The initial velocity is -24
e. none of these
2
11. Give the sign of the 2nd derivative of f at the given point. f ( x) = cos( x) at x =
a. positive
b. negative
12. Differentiate: a. f ( x) = 10 x
c. 0
d. cannot be determined
b.=
f ( x) log 2 ( x 2 + 7)
4
e. none of these
c. f ( x) = x 5e5 x
13. Find the derivative of each at the given point using the table below:
x
f ( x)
f '( x)
g ( x)
g '( x)
1
4
5
4
5
4
0
7
1
½
π
6
6
4
6
3
a. h( x) = f ( g ( x)) at x = 6
b. h( x) = g ( x ) at x = 16
c. h( x) = sin( f ( x)) at x = 4
d. h=
( x) f (2 x + g ( x)) at x = 1
Calculus First Semester Review 2
x2 − x
,
x
b) lim h( x) .
1. Determine the limit of
a) lim h( x)
x →0
h( x ) =
2. Determine the limit:
a) lim
x →−1
x 3 + 125
.
x →−5
x+5
t+2
t −4
b)
t →−2 2
1
lim
( x − 2)
x→2
2
2x2 + x + 1
x+2
3. Determine the limit: lim
4. Determine the limit: lim
5. Determine the intervals of continuity:
6. Determine the value of c such that the function is
f ( x) =
 x + 3, x ≤ 2
cx + 6, x > 2
3x − x − 2
x −1
7. a) lim
x →0
2
sin x
x
continuous for all real numbers: f ( x) = 
b) lim
x →0
sin 2 x
x
9. Find the derivative:
a) f ( x=
) x3 − 3x 2
x →−2−
8. a.) lim
x →∞
x+2
3x − 1
b) lim
x →∞
x
x +1
2
10. Find dy/dx:
x +1
b) f ( x) =
x −1
11. Find the derivative of y = 2sinxcosx
13. Use implicit differentiation to find dy/dx:
x 2 + 3 xy + y 3 =
10
a) =
y
1 − x3
b) y=-xtanx
12. Find the derivative y = ln x
14. Determine where the function is
a) differentiable
b) continuous but not differentiable
c) neither continuous nor differentiable:
2 x − 3, −1 ≤ x < 0
f ( x) = 
 x − 3, 0 ≤ x ≤ 4
15. Find the derivative: y =
sin x
x2
17. Find the derivative: f ( x) =
9
3x 2 − 2 x
16. Find the equations of the tangent line and the
normal (perpendicular) line at the given point:
a) =
b) x 2 + y 2 =
20 at (2, 4)
y ( x + 3) 2 at (-2,1)
2
2
18. Differentiate:
=
y sec x + tan x
19. Suppose the position equation for a moving object 20. Find the point(s) of any horizontal tangents:
is given by s(t ) = 3t 2 − 2t + 5 where s is measured in
x 2 + xy + y 2 =
6
meters and t is measured in seconds. Find the velocity
and acceleration when t = 2.
21. Use the graph of the derivative to find the following:
a. find intervals of increasing and decreasing b. intervals of concavity c. sketch a graph of f.
22. Find the derivative. (Inverse Trig derivatives will not be on the Fall final, but long term you need them.)
a. f ( x) = tan −1 (5 x)
b. f ( x) = cos −1 (3x)
c. f ( x) = csc −1 ( x 2 )
Answers to graphs. The dark graph is the graph of f.