Practice Problems
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Practice Problems |
Printed by: | Guest user |
Date: | Saturday, September 7, 2024, 7:40 PM |
Description
Work through the odd-numbered problems 1-37. Once you have completed the problem set, check your answers.
1. Match the graphs of the three functions in Fig. 8 with the graphs of their derivatives.
In problems 3-5, find the slope of the secant line through the two given points and then calculate
.
7. Use the graph in Fig. 10 to estimate the values of these limits. (It helps to recognize what the limit represents.)
In problems 9 – 11, use the Definition of the derivative to calculate and then evaluate
.
13. Graph and
. Calculate the derivatives of
,
, and
.
In problems 15 – 17, find the slopes and equations of the lines tangent to at the given points.
19. (a) Find the equation of the line tangent to the graph of at the point
.
(b) Find the equation of the line perpendicular to the graph of at
.
(c) Where is the tangent to the graph of horizontal?
(d) Find the equation of the line tangent to the graph of at the point
.
(e) Find the point(s) on the graph of
so the tangent line to the curve at
goes through the point
.
21. (a) Find the angle that the tangent line to at
makes with the x–axis.
(b) Find the angle that the tangent line to at
makes with the x–axis.
(c) The curves and
intersect at the point
. Find the angle of intersection of the two curves (actually the angle between their tangent lines) at the point
.
23. Fig. 13 shows the graph of the height of an object at time . Sketch the graph of the object's upward velocity. What are the units for each axis on the velocity graph?
25. A rock dropped into a deep hole will drop feet in
seconds.
(a) How far into the hole will the rock be after 4 seconds? 5 seconds?
(b) How fast will it be falling at exactly 4 seconds? 5 seconds? seconds?
27. It costs dollars to produce
golf balls. What is the marginal production cost to make a golf ball? What is the marginal production cost when
? when
? (Include units.)
29. Define to be the area bounded by the x–axis, the line
, and a vertical line at
(Fig. 15).
(b) Find a formula which represents for all
?
In problems 31 – 37, find a function which has the given derivative. (Each problem has several correct answers, just find one of them.)
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.2-Definition-of-Derivative.pdf This work is licensed under a Creative Commons Attribution 3.0 License.
(e) DNE
(f) DNE
13. For any constant , if
, then
The graphs of and
are "parallel" parabolas:
is
shifted up 3 units, and
is
shifted
down 5 units.
15. . Then
and the
equation of the tangent line at
is
or
.
and the equation of the tangent line at
is
or
.
17. . Then
and the equation of the tangent line
at
is
or
.
and the equation of the tangent line at
is
or
.
(c) so the tangent line is horizontal when
: at the point
.
(d) (the slope of the tangent
line) so
or
.
Since , the equation of the
tangent line becomes
.
(e) We need such that
. Then
. There are two points with the
property we want:
and
.
21. (a) , so when
. Angle
radians
.
(b) , so when
. Angle
radians
.
(c) Angle radians
radians (or angle
)
23. Graph. On the graph of upward velocity, the units on the horizontal axis are "seconds" and the units on the vertical axis are "feet per second".
25. (a) ft.
ft. (b)
ft/sec
ft/sec.
27. dollars to produce
golf balls.
Marginal production cost is dollars per golf ball.
dollars per golf ball.
dollars per golf ball.
29.
(d) represents the rate at which
is increasing, the rate at which area is accumulating.
31 .