Weekly Tutorial 1 for Math. 69.107A,
Elementary Calculus I

Instructor: Dr. Angelo Mingarelli

Notes: These are group tutorials; each question is to be divided among the group (at most 5 students in a given group), solved and submitted to the T.A. at the end of the 50 min. period. NO NOTES ALLOWED, only group interaction...All students within a group will be assigned the same grade for this tutorial...
This online test is meant to be printed and handed in.
Solutions will appear later on during the term ...

Group names (Please PRINT) & Student Numbers:

1. Calculate the derivative of the following function: f(x) = sqrt(x + x3).
[Here, "sqrt" means the "square root of ..."]

a) 1 + 3x2
b) (1/2)(x + x3)-1/2
c) (1 + 3x2)/(2 sqrt(x + x3))
d) (1/2)(x + x3)-1/2 (3x2)


2. Find the equation of the tangent line to the curve

x2y + y2x - xy = 4

at the point (2,1)

a) 3x - 2y = 7
b) 2x + 3y = 7
c) 3x + 2y = 7
d) 2x - 3y = 7


3. A twice differentiable function f has the property that f '(x) + f(x) = 0 (i.e., the derivative of the function plus the function itself always equals zero, for each real number x), and f(0) = 1.
Which of the following statements is true?

a) The values of f '(x) > 0 for every real number x
b) The function f is concave down (i.e., f "(x) < 0 ) for every real number x
c) There is an interval, I, to the right of x = 0 on which f '(x) < 0 (for every x in I)
d)There exists a real number x > 0 at which the graph of f has a horizontal tangent


4. Calculate the derivative of the following function: g(x) = (sqrt(x) + 1) / (sqrt(x) - 1)

a) (2 sqrt(x) + 1) / (2 sqrt(x) - 1)
b) -1
c) (1/2)(sqrt(x) - 1)-1/2
d) -x-1/2 (sqrt(x) - 1)-2


5. Find the slope of the tangent line to the curve

sqrt(xy) = x - 2y

at the point (4,1)

a) 1/4
b) 4
c) -1/4
d) 1


6.You are given the graph of a polynomial function, f(x), as follows:

Consider the graph of the derivative of this function ... How many zeros does the derivative have (i.e., how many solutions does the equation f '(x) = 0 have) ?

a) none
b) 5
c) 6
d) 7


7. Refer to question #6, above. On which of the following intervals is the derivative function, f '(x) > 0 or positive?

a) On (8,10)
b) On (11,13)
c) On (10,12)
d) On (0,5)

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