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This set of problems is for preparing for the final exam of Math 190.
You need to do more problems in order to perform well on the test.
1. Use a graph to find a number δ such that sin x
1
0.2 whenever x .
6
2
2. If 1 f ( x) x 2 8x 8 for all x, find lim f ( x) .
x1
3. (a) Use the definition of derivative to find the slope of the tangent line to the
graph of the exponential function y 8 x at the point (0,1).
(b) Estimate the slope to three decimal places.
4. Use the midpoint rule with n = 4 to approximate the area of the region bounded by
the given curves. y 1 x 3 , y 1 2 x , x 2 .
5. Find the area of the region bounded by the parabola y x 2 , the tangent line to this
parabola at (6,36), and the x-axis.
5
6. Evaluate the integral.
4x x
2
dx
2
x
t2
1
1
7. If F ( x) f (t ) dt , where f (t )
10 u 4
du , find F " (2).
u
8. Find the inflection point for the function. f ( x) 6 5 x 2 sin x, 0 x 3 .
9. Use Newton’s method with the initial approximation x1 =2 to find x 4 , the fourth
approximation to the root of the equation x 4 13 0 .(Give your answer to four
decimal places.)
10. Consider the figure below, where AC = 7 ft, BD = 1ft and AB = 6 ft. How far
from the point A should the point P be chosen on the line segment AB so as to
maximize the angle θ? Round you answer to the nearest hundredth.
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11. Evaluate the limit. lim
7 x 1 7 1
x
x 0
12. Use implicit differentiation to find an equation of the tangent line to the curve
4 x 2 3 y 2 7 at the point (1,1).
13. Find the limit. lim 1 10 x
1/ x
x0
14. Find the exact values of the numbers c that satisfy the conclusion of The Mean
Value Theorem for the function f ( x) x 3 5 x for the interval [-5,5].
1
15. Find
tan(
2
) d .
0
16. The velocity of a particle is v(t ) t 3 10t 2 24t
ft / s . Compute the
(a) displacement over [0,4], [4,6], and [0,6].
(b) Total distance traveled over [0,6].
17. Find lim
x1
1 x2
.
cos 1 x
18. It costs you c dollars each to manufacture and distribute backpacks. If the
backpacks sell at x dollars each, the number sold is given by
a
n
b(100 x) where a and b are positive constants. What selling price
xc
will bring a maximum profit?
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