Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 40
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411
1
π/2
ln(u)
(a)
du
(c)
t dt
(e)
t dt
u
1/2
1/2
π/6
π/2
ln(π/2)
ln(u)
(b)
du
(d)
t dt
u
π/6
ln(π/6)
sin(x)
12. Compute the indefinite integral
dx.
2
1 + cos
(x)
x
2
13. Let F (x) =
sin
(t) dt. Evaluate the limit
0
F (x)
lim
.
2
x
x
0
x
d
sin(t)
5
14. Evaluate
(x
dt).
dx
t
2
15. Compute the following limits.
x
9
t
e
(a) lim
(c) lim
2
x
3
t
t
9
x
2
sin(3θ)
(ln x)
(b) lim
(d) lim
10θ
x
θ
0
x
16. (a) State the limit definition of the continuity of a function f at x = a.
(b) State the limit definition of the derivative of a function f at x = a.
2
x
if x < 1
(c) Given f (x) =
. Is the function continuous at x = 1? Is the function dif-
4
3x if x
1
ferentiable at x = 1? Use the definition of the derivative. Graph the function to check your
answer.
17. Provide the most general antiderivative of the following functions:
4
2
(a) f (x) = x
+ x
+ x + 1000
20
(b) g(x) = (3x
2)
sin(ln(x))
(c) h(x) =
x
dy
18. Use implicit differentiation to find
, and compute the slope of the tangent line at (1,2) for the
dx
following curves:
2
2
(a) x
+ xy + y
+ 9x = 16
2
2
(b) x
+ 2xy
y
+ x = 2
19. An rock is thrown up the in the air and returns to the ground 4 seconds later. What is the initial
velocity? What is the maximum height of the rock? Assume that the rock’s motion is determined by
2
the acceleration of gravity, 9.8 meters/second
.
20. A conical tank with radius 5 meters and height 10 meters is being filled with water at a rate of 3 cubic
meters per minute. How fast is the water level increasing when the water’s depth is 3 meters?
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