Exam Calculus I: Limits and Derivatives - May 26, 2009, 1.00, Supreme Islamic Council 1-4
The final mark of Calculus I is based on the Mark Basic Math (30%), Assignment (10%) and the Exam Limits
and Derivatives (60%). The assignment is only counted (10%) if the mark is greater than the final mark.
First try to solve the problems on a scrap paper and then write down your answer (or use a pencil and
eraser). The use of a calculator is allowed, but not necessary.
Exercise 1: Find the following limits, if they exist (include ∞ and -∞ as answers).
If a limit does not exist, state why not.
Exercise 2: Find the derivative of the following functions:
Exercise 3: Given is the function
a. State the derivative of .
b. Give the equation of the tangent line at the point (2, ½ ) of the graph of f .
c. Proof that the answer of part a. is correct, using the definition of derivative.
Exercise 4: Given is the function
a. State the Domain of F.
b. At what values of x is F continuous and differentiable? Explain you answer.
c. Compute for each value a, at which F is discontinuous, the limit from the left,
, and the limit
from the right,
.
d. Give for each a the type of discontinuity (removable, jump, asymptotic). Use the results of part c.
Exercise 5: Given is the function
a. Find
(include ∞ and -∞ as answers).
b. Which rule should you use to find the derivative
? Give the formula and use it to check that
c. Find the absolute minimum and the absolute maximum of
on the interval [0, 2]
Exercise 6 Investigate the function
:
a. State the Domain of f.
b. Find the y-intercept and the x-intercept(s) of f .
c. Does the function
have special properties (is it even/odd/periodic?). Motivate your answer.
d. Compute
and give the equations of all asymptote(s).
e. Compute the first and the second derivative of f(x).
f. Find all local minima and maxima. Which of these extremes is an absolute maximum or minimum?
g. Sketch the graph of f.
--------------Success!--------------Marks: mark exam = total/7 (rounded at one decimal) . Mark Calculus I = 0.3Test + 0.1Assignment + 0.6Exam
1
2
3
4
5
6
Total
a b c d e f g h a b c d a b c a b c d a b c a b c d e f g
2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 2 2 4 2 2 2 4 2 4 2 2 2 4 3 70
Solutions: Exam Calculus I: Limits and Derivatives - May 26, 2009
Exercise 1
Exercise 2:
Exercise 3 a.
b.
=> the equation of the tangent line at (2, ½ ) is y - ½
(x - 2) or y
x+1
Exercise 4
a. Domain of F :
=> DF = {x| x  5 and x -3} (or DF = (-∞, -3)  (-3, 5)  (5, ∞) )
b. F is continuous and differentiable at all values of the domain, because F is a rational function.
d. (see results at c.) F has a asymptotic discontinuity at -3 and a removable discontinuity at 5.
Exercise 5
c.
x = 3 is outside the closed interval: x = 1 and the interval boundaries have the following function values,
according to the closed interval method (g is continuous and differentiable on [0, 2]):
f(1) = e4
f(0) = e0 = 1
f(2) = e2
=> the largest value is f(1): absolute maximum on [0, 2]
and the smallest value is f(0): absolute minimum on [0, 2]
Exercise 6
a. Df =
b. y-intercept: f(0 ) = 0. x-intercepts:
c.
.
There are no horizontal or vertical asymptotes
f.
Because - at x = -2
changes from + (increasing) to – (decreasing), f(-2) = 16 is a local maximum
- at x = 2
changes from + (increasing) to – (decreasing), f(2) = 16 is a local maximum
- at x = 0
changes from – (decreasing) to + (increasing), f(0) = 0 is a local minimum
The sign scheme of
:
+ + + 0 - - - -
0 + + + + 0 - - -
-2
0
2
Because of the local extremes found, the absence of
vertical asymptotes and the limits at infinity (see d),
there is no absolute minimum and there are two
absolute maxima f(-2) = 16 and f(2) = 16.
g.
The graph of f :