Calculus, Unit 2: Review_teacher, page 1
1. Find the instantaneous rate of change of f(x) = 7x2 – 3x at x = 2.
2. Find the average velocity of the time-position function s(t) =7t2 – 3t meters
between t = 5 sec and t = 8 sec.
3. What is the instantaneous velocity at t = 5 sec of the time-position function s(t)
= t3 – 5 meters?
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Calculus, Unit 2: Review_teacher, page 2
4. Find the equation of the tangent line to the curve given by f(x) = x2 + 7x – 17 at
x = –5.
5. What is the equation of the normal line at x = –5 of the curve given by f(x) =
x2 + 7x – 17?
6. Find the equation of the normal line at x = 1 of the function f(x) = x + 3.
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Calculus, Unit 2: Review_teacher, page 3
7. If y = 3x2 + 5x – 1 what is
?
8. Find the derivative of f(x) = x + 2
3 at x = 7.
9. Use the formal definition of the derivative to find the slope of the normal line
to the curve f(x) = 1/(x + 1) at x = –4.
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Calculus, Unit 2: Review_teacher, page 4
10. Find the function for the velocity where the time-position function is given by
s(t) = t – t2 feet. (t is given in minutes).
11. The left picture is the
function f. Sketch its
derivative f’ on the right
coordinate system.
12. The left picture is the
function f’. Sketch the
original function f on the
right coordinate system.
13. Separate the following six items into two associated groups of three items
each: Increasing function, Decreasing function, Negative slope, Positive slope,
Positive derivative, Negative derivative.
Increasing function
Decreasing function
Positive slope
Negative slope
Positive derivative
Negative derivative
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Calculus, Unit 2: Review_teacher, page 5
14. Using the function whose graph is shown to the
right, specify the following intervals:
a. Interval(s) of negative derivative
(–ϱ͕ϬͿ͕;ϰ͕ьͿ
b. Interval(s) of positive slope
(–ь͕–5), (0, 4)
c. Interval(s) of decrease
(–ϱ͕ϬͿ͕;ϰ͕ьͿ
In problems 15-18͕ determine the point(s) at which the function is not
differentiable. State the reason why.
15.
16.
x = 6, discontinuity
17.
x = –6, disc; x = 5, vert tan
18.
x = –3, cusp
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Differentiable everywhere
Calculus, Unit 2: Review_teacher, page 6
19. Determine by an analysis of a continuity test and “left” & “right” derivatives if
this function is differentiable at x = –2.
f(x) = 5x
20x
if x
20 if x >
2
2
a. The point(s) at which the tangent line is
20. Use the letters associated
horizontal.
with the points on this function
C, D
to answer the questions.
b. The point(s) at which the function has a root.
B
c. The point(s) at which the function value is
negative and the derivative is positive.
A
d. The point(s) at which the function value is
positive and the slope of the tangent line is
positive.
E
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