Download Math 180 #003 U M Summer 2012 Elements of Calculus I

Math 180 #003
UM
Summer 2012
Elements of Calculus I: Worksheet 3
1. Find the equation of the tangent line to the graph of f ( x) =
1
at x = 1 .
x2
2. Let f ( x) = 1 / x . Find the area of the triangle formed by the x-axis, y-axis, and the tangent
line to the graph of f ( x) at the point x = 2 .
3. Find the x-coordinate of the point on the graph of f ( x) = x 2 where the tangent line is
parallel to the segment joining the points on the graph where x = −2 and x = −1 .
4. The tangent line to the curve f ( x) = 13 x3 − 4 x 2 + 18 x + 22 is parallel to the line
6 x − 2 y = 1 at two points on the curve. Find the x-coordinates of the two points.
5. The line 2x + y = b is tangent to the graph of f ( x) = ax 2 at x = 2 . Find a and b.
6. Find the equation of the line that is perpendicular to the tangent line of f ( x) = x3 − 3 x + 1
at the point (2,3) and passes through that point.
7. In the figure at the right, the line y = 14 x + b is
y = 14 x + b
tangent to the graph of f ( x) = x at x = a . Find the
values of a and b.
f ( x) = x
8. Find all the coordinates on the graph of
f ( x) = (2 x 4 + 1)( x − 5) where the slope is 1.
9. Suppose that f (3) = 1 , f '(3) = 2 , g (3) = −2 , and
g '(3) = −1 . Let h( x) = f ( x) g ( x) . Find the equation
of the tangent line to the graph of h( x) at x = 3 .
10. Suppose that f (1) = 2 , f '(1) = 3 , g (1) = 4 , and g '(1) = 5 . Evaluate
11. Given f ( x) , let g ( x) = x 2 f ( x) and h( x) =
(a) Find g '( x) .
12. Given f ( x) , g ( x) , and h( x) , find
f ( x)
.
x3
(b) Find h '( x) .
d
[ f ( x ) g ( x ) h( x ) ] .
dx
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a
d  f ( x) 
.
dx  g ( x)  x =1
13. Given f ( x) , g ( x) , and h( x) , find
14. Suppose that
d  f ( x) g ( x) 
.
dx  h( x) 
d
 f ( g ( x) )  = 3 x 2 i f '( x 3 + 1) . Find g ( x) .
dx
15. Find the derivative of the following functions:
6
4
(a) y = 3 x 2 + ( 6 x − 7 x 3 )  .


(b) y =
5 x3 − 3x 2 +
6
.
x
16. Find the equation of the tangent line to the curve f ( x) = ( x − 2)5 ( x + 1) 2 at x = 3 .
17. Given f (a ) = 1 , g (a ) = 4 , f '(a ) = −1 , and g '(a ) = 4 , let h( x) = [ f ( x) ] + g ( x) .
2
(a) Evaluate h(a ) .
(b) Evaluate h '(a ) .
2
2
 f ( x) 
18. Given f ( x) , let g ( x) = x 2 [ f ( x)] and h( x) =  3  .
 x 
(a) Find g '( x) .
(b) Find h '( x) .
19. Suppose that f (1) = 2 , f '(1) = 3 , f '(5) = 4 , g (1) = 5 , g '(1) = 6 , g '(2) = 7 and g '(5) = 8 .
d
d
(a) Evaluate
 f ( g ( x) )  .
(b) Evaluate
 g ( f ( x) )  .
dx
dx 
x =1
x =1
20. Let f ( x) = 1 / x and g ( x) = 1/ (2 − x) . Find the point where these graphs intersect and
prove that they intersect at right angles.
21. Consider the graph of r ( x) to the right. Evaluate:
d 
r ( x) − 2 x 
.
dx 
x =−3
22. Classify the following as true or false:
(a) If f ( x) = π x , then f '( x) = xπ x −1 .
d
(b)
[ f ( x) g ( x)] = f '( x) g '( x) .
dx
d  f ( x)  f '( x)
(c)
=
.
dx  g ( x)  g '( x)
d
(d)
 f ( g ( x) )  = f ' ( g '( x) ) .
dx 
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r(x)