MATH 121 Final: MATH 121 Amherst S11M12 2802Starr 29Final 0
15 Feb 2019
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Math 105 Final Exam
1. Evaluate each of the following limits. Justify your answers completely and be clear whether
the limit equals a value, +∞or −∞, or does not exist.
(a) [5]lim
x!2
x2+2x−3
x2+x−2
(b) [5]lim
x!1
x2+3x−4
|1−x|
(c) [5]lim
x!2+
x2−3x+2
x2−4x+4
(d) [5]lim
x!7
7
x−1
x+6
x+7
(e) [5]lim
x!3−
x2−8x+ 15
1−8x+g(x+ 1),whereg(x)=x2+ 7.
(f) [5]lim
x!3
x2−4x−21
√1−x−2
(g) [5]lim
x!1
√x2+1
2x−3
(h) [5]lim
x!1
x2−7x+4
x4+x2−3
2. Compute each of the following derivatives.
(a) [5]f0(1), where f(x)= x2+1
x√x+2x+1.
(b) [5]g00(x), where g(x)= x
2−x
(c) [5]
dy
dx,wherex2y3+3x3−2y2= 26.
(d) [5]g0(x), where g(x)=✓1
x3+7x◆4✓x4−1
x7◆5
.
3. Let f(x)=√3x−2.
(a) [10]Calculate f0(x), using the limit definition of the derivative.
(b) [5]Check your answer using the Chain Rule.
4. [10]Consider the equation 2y+y2−y5=x4−2x3+2x2. Find the equation of the tangent line to
this curve at the point (1,0).
5. [15]Find the absolute maximum and absolute minimum values of
f(x)=x2√5−xon [1,5].
6. [20]Let f(x)=x+2
x3.Take my word for it that:
f0(x)=−2(x+ 3)
x4and f00(x)=6(x+ 4)
x5.
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