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Math 261 (Calculus I)
Chapter 4 practice test
Name:
1. Find abs. max and abs. min of the function in the given domain: f ( x)  x  2cos x,    x  
……………………………………………………………………………………………….
4
2. Show that the equation x  4 x  c  0 has at most two real roots
……………………………………………………………………………………………….
3. Find a function f such that
f ( x)  x3 and the line x  y  0 is tangent to the graph of f .
……………………………………………………………………………………………….
4. Use the Newton’s Method to find the third approximation to the root of the given equation. (Give
your answer to two decimal places.)
x3  x 2  1  0, x1  1
……………………………………………………………………………………………….
x2 y 2
5. Find the area of the largest rectangle that can be inscribed in the ellipse

 1.
4 9
……………………………………………………………………………………………….
6. If f (1)  10 and f ( x)  2 for 1  x  4, how small can f(4) possibly be?
……………………………………………………………………………………………….
For problem 7, use the guidelines of the section 4.5, to sketch the following curve.
7.
y
1
x  sin x, 0  x  3
2
……………………………………………………………………………………………….
8. Find abs. max and abs. min of the function in the given domain:
f ( x)  (cos x) /(2  sin x), 0  x  2
……………………………………………………………………………………………….
9. If f (1)  10 and f ( x)  2 for 1  x  4, how small can f(4) possibly be?
……………………………………………………………………………………………….
10. Show that the equation 2 x  1  sin x  0 has exactly one real root.
……………………………………………………………………………………………….
12. A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration
of 22
ft / s 2 . What is the distance covered before the car comes to a stop? (Use the fact 50mi/h =
220/3 ft/s)
……………………………………………………………………………………………….
For problem 13 and 14, use the guidelines of the section 4.5, to sketch the following curve.
13.
y
x2
x2  9
……………………………………………………………………………………………….
14. f (t )  t  cos t ,  2  t  2
……………………………………………………………………………………………….
15. If the graph of a cubic function has three x-intercepts x1 , x2, and x3, show the x-coordinate of the
inflection point is
 x1  x2  x3  / 3.
……………………………………………………………………………………………….
16. Sketch the graph of a function that satisfies all of the given conditions.
f   2   0, f  2   1, f  0   0
f   0   0 if 0  x  2, f   x   0 if x  2,
f   x   0 if 0  x  1 or if x  4,
f   x   0 if 1  x  4, lim x  f  x   1,
f   x   f  x  for all x
……………………………………………………………………………………………….
Use the guidelines of the section 4.5, to sketch the following curve.
17.
y  x 4  4 x3
18. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of
lengths 3cm and 8 cm if two sides of the rectangle lie along the legs.
19. A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest
possible volume of such a cylinder.
20. Find the equation of the line through the point (3, 5) that cuts off the least area from the first
quadrant.
21. Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other
two vertices above the x-axis and lying on the parabola.
y = 5 − x2