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Calculus
Name:
Chapter 2a Test, Part I – Graphing Calculators Required
2015-2016
Exercises 1-5, are multiple choice; choose the best answer. (3 points each)
1
1. If f (x) = −x3 + x + , then f 0 (−1) =
x
(a) 3
(b) 1
(c) −1
(d) −3
(e) −5
2. Assume f 0 (c) = −4. Determine f 0 (−c) if f is an even function.
(a) 0
3. Find
(b) −4
(c) −2
(e) −8
(d) 4
dy
for y = 4 sin2 (3x)
dx
(a) 8 sin (3x)
(b) 24 sin (3x)
(d) 12 sin (3x) cos (3x)
(e) 24 sin (3x) cos (3x)
(c) 8 sin (3x) cos (3x)
4. Which of the following is an equation for the line tangent to the graph of
(a) 13x − y = 8
2x + 3
at the point (1, 5)?
3x − 2
(b) 13x + y = 18 (c) x − 13y = 64 (d) x + 13y = 66 (e) −2x + 3y = 13
5. Let f (x) = (2x + 1)4 . Determine the value of the 4th derivative at the point x = 0.
(a) 0
(b) 24
(c) 48
(d) 240
(e) 384
For exercises 6- 8, completely answer each question.
1
. Use the limit definition of a derivative to find f 0 (x). [5 points - 1 point will be
2x + 1
awarded for the derivative itself. The remaining 4 points will be awarded for the correct use and
simplification of the limit definition.]
6. Let f (x) =
7. Find the equation of the line tangent to f (x) = (x − 5) (x2 − 4) at the point (3, −10) [3 points]
8. An object is thrown (up) from the top of a 220-foot building with an initial velocity of 26 feet
per second. Thus, its position function is s(t) = −16t2 + 26t + 220, where s is the object’s height
(measured in feet) and t is the amount of time since the object’s release (measured in seconds.)
(a) What is the velocity of the object the moment it hits the ground? [3 points]
(b) What is the acceleration of the object the moment it hits the ground? [3 points]
Calculus
Name:
Chapter 2a Test, Part II – No Calculators
2015-2016
Exercises 9- 13, are multiple choice; choose the best answer. (3 points each)
9. Determine the point(s) on the graph of the function f (x) = x3 − 2 where the slope is 3.
√ (a) There are none.
(b) 3 2, 0
(c) (1, −1), (−1, −3)
(d) (1, 3)
(e) (1, 3), (−1, 3)
10. If y = tan x − cot x, then
(a) sec x csc x
dy
=
dx
(b) sec x − csc x
(c) sec x + csc x
(d) sec2 x − csc2 x (e) sec2 x + csc2 x
11. If y = cos2 x − sin2 x, then y 0 =
(a) −1
(b) 0
(d) −2 (cos x + sin x)
(e) 2 (cos x − sin x)
3
, then f 0 (x) =
4 + x2
−6x
3x
(a)
(b)
2
2
(4 + x )
(4 + x2 )2
(c) −2 sin (2x)
12. If f (x) =
(c)
6x
(4 + x2 )2
−3
(4 + x2 )2
(e)
3
2x
−x + 3
(d) √
2x − 3
(e)
5x − 6
√
2 2x − 3
(d)
√
13. If f (x) = x 2x − 3, then f 0 (x) =
3x − 3
(a) √
2x − 3
(b) √
x
2x − 3
(c) √
1
2x − 3
For exercises 14- 16, completely answer each question.
14. Let f (x) = x3 + 2x2 + 6x. Determine all points on the graph that have a horizontal tangent line. [3
points]
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15. Let f (x) = (x2 − 5) . Determine both f 0 (x) and f 00 (x). [3 points each]
16. Create a piecewise function with 3 different pieces that is continuous at every point, but not
differentiable at exactly one x-value. Additionally, identify the x-value at which your function is
non-differentiable and explain why. [5 points]
17. Consider the graph below, showing y = f (x) and y = g(x). Both f and g have a domain of [−3, 9].
9
y
y = f (x)
y = g(x)
x
−3
9
−1
(a) Calculate (g ◦ g)0 (0). (3 points)
f
(b) On what values is
not differentiable on the interval [−3, 9]? (3 points)
g