MATHS

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Rossella Bartolillo F2010MATH249L06
Assignment 5 is due : 10/25/2010 at 11:59pm MDT.
The number of attempts available for each question is noted beside the question. If you are having trouble figuring out your error,
you should consult the textbook, or ask a fellow student, one of the TA’s or your professor for help.
There are also other resources at your disposal, such as the Engineering Drop in Centre and the Mathematics Continuous Tutorials.
Don’t spend a lot of time guessing – it’s not very efficient or effective.
Make sure to give lots of significant digits for (floating point) numerical answers. For most problems when entering numerical
answers, you can if you wish enter elementary expressions such as 2∧3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧(ln(2)) instead
of 2, (2+tan(3)) ∗ (4−sin(5))∧6−7/8 instead of 27620.3413, etc.
1. (1 pt) In this question, we shall take steps to find the values
of a and b , given that the function
f(x) =
x
2 +4x+7 if x ≤ 2
ax+b if x > 2
is differentiable at 2 .
(a) It is known that if a function is differentiable at a point
c, then it is continuous at c. Using now the continuity of f at
2, we can establish a relationship between a and b. Find this
relationship and express it in the form b = Aa+B, where A and
B are constants.
Answer: b = a + .
(b) Assuming that x > 2, one can simplify the quotient
f(x)− f(2)
x−(2)
into the form Ca + D, where C and D are constants. Find
these constants.
Answer: C = , and D = .
Hint. Don’t forget that you can use the result from part (a)
to eliminate b from your expression.
(c) Assuming that x < 2, one can simplify the quotient
f(x)− f(2)
x−(2)
into the form Ex + F, where E and F are constants. Find
these constants.
Answer: E = , and F = .
(d) Using the results of parts (a), (b) and (c), find the values
of a and b.
Answer: a = and b = .
2. (1 pt)
Identify the graphs A (blue), B( red) and C (green) as the
graphs of a function and its derivatives:
is the graph of the function
is the graph of the function’s first derivative
is the graph of the function’s second derivative
3. (1 pt)
Differentiate the following functions and simplify your answers into the specified form.
(a) f(x) = −7x−3
3x
2 −10
.
1
Answer: f
0
(x) = 1
(3x
2 −10)
2
( x
2+ x+ ) [Note:
The three answers you enter should be constants.]
(b) g(x) = x
3
12x+10
.
Answer: g
0
(x) = 1
(12x+10)
2
( x
3+ x
2+ x+
) [Note: The four answers you enter should be constants.]
4. (1 pt) Differentiate the following functions and simplify
your answers into the specified form.
(a) f(x) = (12x−4)sinx+ (−9x−1)cos x.
Answer: f
0
(x) = ( x+ )sinx + ( x+ )cos x.
[Note: The four answers you enterr should be constants.]
(b) g(x) = (4x
2 +7)tanx.
Answer: g
0
(x) = ( )sec2
x+ ( )tanx.
[Note: The two answers you enter should be polynomials in x.]
5. (1 pt) Find g
0
(4) given that f(4) = −3 and f
0
(4) = 5
(a) For g(x) = 3x
2 −5 f(x)
g
0
(4) =
(b) For g(x) = 2x+1
f(x)
g
0
(4) =