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Material Type: Quiz; Class: Calculus I; Subject: Mathematics; University: Millikin University; Term: Spring 2001;
Typology: Quizzes
1 / 3
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2/8/09 Major Quiz 2 Review Problems
1.) Find dx
dy for each of the following.
(a) y = 3x
3 − 5x
2
5
4 ) (c) 2
3 3
(^5) x
x y = +
(d) 4 5
x x
y = − (e) 2
1 2
1
− y = x − x (f) 1
2
x
x x y
(g) y = (4x
2 − 1)(2x
3
3
3
x
x y (i) y = (2x
4 − 3x)(5x
2 − 2x + 5)
(j) 6
3
2
x
x y (k) y = 2 5
x
x (l) y = 2
3
4
5
4
x
x
x
x − + −
2.) Use the definition of derivative to find f'(x) for f(x) = 3x
2
3.) Prove the following theorem.
4.) Find an equation of each of the lines through the point (1, 2) that is tangent to the curve
y = 2x
2
5.) Find the values of a and b such that f is differentiable at 2 if f(x) =
2 x if x
ax b if x .
6.) Find an equation of the tangent line and normal line to the curve y = 3x
4 − 12x at the
point (1, −9).
7.) Use the definition of derivative to find f'(x) for f(x) = 2x
2 − 3x + 1.
(a) f(x) = 5x
3
2 − 3x (b) f(x) = (3x + 5)(2x
2 − 3x +1) (c) f(x) = 3 7
5
x
x .
(d) f(x) = 4 2
x x
− (e) f(x) = 2 4
2
x
x
2
10.) Use the Intermediate Value Theorem to determine if
2 f ( ) x = x − 29 has a zero in the interval
[5,8].
11.) Find the derivative of
2 –9 + + 7 ( )
x x f x x
12.) Using the position function
s t ( ) – 9 t
= , find the acceleration function.
13.) Find the third derivative of
f ( ) x 3 x + 7 x – x
2/8/09 Major Quiz 2 Review Problems
14.) Below is a graph of f ( ) x. Sketch a graph of f ′( ) x.
3
0
lim h
h h
→ h
equals f ′( ) a for some function f ( ) x and some constant a.
Determine what f ( ) x and the constant a could be.
16.) Use the Intermediate Value Theorem to determine if
3 f ( ) x = x + x − 37 has a zero in the
interval [3, 9].
17.) Using the position function
s t ( ) 4 t – t
= , find the velocity function.
18.) Below is a graph of f ( ) x. Sketch a graph of f ′( ) x.