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MATH 12002 SAMPLE FINAL EXAM ANSWERS
- For each of the following functions, find its derivative with respect to x. Do not simplify
your answers. An answer (even if it is correct) with too much simplification
will receive zero credit in the test.
(a). f (x) = x 4 − 3 x 2
Answer: f ′(x) = 4x^3 − 6 x.
(b). g(x) =
4 x^2 + 1 − sin(2x),
Answer: g′(x) =
8 x−2 cos(2x) 2
√ 4 x^2 +1−sin 2x
(c). h(x) = ln(tan x),
Answer: h′(x) = sec^2 x tan x.
(d). k(x) = (x+1)
1 (^2) (x+2) 1 (^3) (x+3) 1 (^4) (x+4) 1 5 2 ex^2 − 1
Answer: k ′ (x) =
(x+1)
1 (^2) (x+2) 1 (^3) (x+3) 1 (^4) (x+4) 1 5 2 ex^2 − 1
1 2(x+1) +^
1 3(x+2) +^
1 4(x+3) +^
1 5(x+4) −^
2 ex 2 · 2 x 2 ex^2 − 1
(e). f (x) = x 2 e (ln(x^2 +1))^2 ,
Answer: f ′ (x) = 2xe (ln(x^2 +1))^2
- x 2 · e (ln(x^2 +1))^2 · 2 ln(x 2
- · 2 x x^2 +.
(f). g(x) = arcsin(ex),
Answer: g′(x) = e
x √ 1 −e^2 x^
(g). h(x) = ln(arctan x + x),
Answer: h′(x) =
1 1+x^2 + arctan x+x.
(h). k(x) = arctan(x 2 ) + arctan(x − 2 ).
Answer: k ′ (x) = 2 x 1+x^4 +^
− 2 x−^3 1+x−^4 = 0.
(i). f (x) =
∫ (^) ln x ex^ cos
tdt.
Answer: f ′(x) = cos
ln x · 1 x −^ cos^
ex^ · ex.
- Find the equation of the line tangent to the curve
x 2 y − sin(2y) + x 3 = 1
at the point ( 1 , 0 ).
Answer: y = 3x − 3.
- Find the limit if it exists.
(a). lim x→ 1
x^2 − 5 x + 4
x^2 − 1
(b). lim x→+∞
5 x^2 − 100 x
1 − x^2
(c). lim x→ 0
cos(2x) − 1
x^2
(d). lim x→0+
tan x ln x
Answers: (a). −^32 ; (b). − 5 ; (c). − 2 ; (d). 0.
- Find the average value of f (x) = x 2
x^3 + 1 on [0, 2].
Answer: 26
If the radius of a circle is increasing at a constant rate of 5cm/sec, how fast is the area
of the circle changing when the radius is 2 cm?
Answer: 20 π cm 2 /s.
- Given f (x) = x 3 − x 2 − x − 1 ,
(a). find (the x-coordinates of) the critical points of f (x);
Answers: x = −^13 , 1
(b). determine the intervals where f (x) is increasing or decreasing;
Answers: increasing on (−∞, − 1 /3) and (1, +∞), decreasing on (− 1 / 3 , 1).
(c). determine the intervals where f (x) is concave up or concave down;
Answers: concave up on (1/ 3 , +∞), concave down on (−∞, 1 /3).
(d). determine the x-coordinates of the local maximum, local minimum and inflection
points;
Answers: local max. at x = − 1 / 3 , local min at x = 1, inflection point at x = 1
(e). find the global maximum and global minimum of f (x) on the interval [0, 2].
