Math 229 Sample Final Exam
1. [12] Give the definition of the derivative f0(x) of a function f(x), and use it to find
(1
x)0.
2. [24] The position s(t) of a point moving along a straight line is given by
s(t) = 10t
t2+ 1 t≥0.
Given that s0(t) = 10 1−t2
(t2+1)2and s00(t) = 20 t(t2−3)
(t2+1)3answer the following:
(a) When does s(t) reach its maximum and what is the value of this maximum?
(b) When is the velocity a maximum and what is the value of this maximum?
(c) What is the average velocity over the time interval [0,2]?
(d) What is the average acceleration over the time interval [0,2]?
3. [12] Find the equations of the tangent and the normal lines to the curve y=x3−2xat
the point (0,0).
4. [20] Evaluate the following limits. [Justify your answers.] (a) lim
x→2
√x−1−1
x−2
(b) lim
x→π
6
sin 3x
2x(c) lim
x→∞
3−2x2
1 + 7x2(d) lim
x→0f(x) where f(x) = sin x
xfor x > 0
1−xfor x≤0.
5. [10] Use calculus to find the point on the line y= 2x−3 closest to the point (5,−3).
6. [25] Find the following: (a) R3
1(x−2+x2)dx (b) Rx(1 −x2)1/2dx
(c) Rsin xcos(cos(x))dx (d) Zπ/4
π/6
tan2x dx (e) Zsin x
cos2xdx
7. [10] Find the area in the first quadrant between the curves y= sin xand y=2
πx. [You
may use your calculator to help find where the curves intersect.]
8. [5] Given a polynomial f(x). Proves that between any two consectutive roots of f(x)
there is a root of f0(x). [Hint: Use Rolle’s theorem.]
9. [24] Differentiate the following: (a) f(x) = cot(√x) (b) f(x)=(x2+ 1)5(x3−1)7
(Do not simplify.) (c) f(x) = x2
x3+1 (d) f(x) = Rx
0sin(t2)dt.
10. [6] Find dy
dx : sin(xy) + x2y3= 1.
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