1. Consider the functionf(x) = sinx.

(a) Verify thatfsatisfies the hypotheses of Rolle's theorem on the interval [0, ]. (b) Find allcthat satisfy the conclusion of Rolle's theorem.

2. Consider the functionf(x) =x3x24x+ 11.

(a) Verify thatfsatisfies the hypotheses of Rolle's theorem on the interval [2,2]. (b) Find allcthat satisfy the conclusion of Rolle's theorem.

1. Verify that the functionf(x) = 2x23x+ 1 satisfies the hypotheses of the Mean Value Theorem on the interval [0,2], and find all numberscthat satisfy the conclusion of that theorem.
2. Supposefis continuous on [2,7],f(2) = 10, and that5f(x)5 for allxin (2,y). Estimatef(7), i.e. find how small and how bigf(7) can be.
3. Prove that the equation
4. has exactly one real solution.

2x3+9x2+42x5 = 0

7. Letf(x) = 3x4+ 4x312x210. Find the (absolute) extremum values offin the

interval [3,2].

8. Sketch a graph of the function

f(x)=|x32x2+x|

The graph should correctly indicatexandyintercepts, local extrema, points of inflection, the intervals wherefis increasing or decreasing, and the intervals wherefis concave upwards or downwards.

Hint.First sketch the graph ofy=x32x2+x.

9. For each of the following functions: (a)f(x)=sinxx,on[2,2] (b)f(x)=x3x

(c)f(x) =x24x21

(d)f(x) =x2+ 4x24

(e)f(x) = cscx

(f)f(x)=1cosx

(g)f(x)=x

sinx4 + 6x

sketch a graph of the function. The graph should correctly indicatexandyintercepts, local extrema, points of inflection, the intervals wherefis increasing or decreasing, the intervals wherefis concave upwards or downwards, and any horizontal or vertical asymptotes.

10. Letf(x)=x45x2+4.

(a) Sketch a qualitatively accurate graph ofy=f(x). (b) Sketch a qualitatively accurate graph ofy=|f(x)|.

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(c) Which one is largerf(1.4562243239684839165) orf(1.4562243239684839167)? (The two arguments are equal except the very last decimal digit.)

1. Find the point of the parabolay=x2+ 3 that is closest to the point (1,2).
2. If 16 square centimeters of material is available to make a box with square base what is
3. the largest possible volume of the box?
4. Consider a right triangle with legs of length 1 and 2, and a rectangle inscribed inside it in such a way that its sides are parallel to the legs of the triangle, as shown below. Find the dimensions of the rectangle so that its area is the maximum possible.
5. 1
6. Use Newton's method to estimate62 starting withx0= 1. Use three iterations.
7. Find the antiderivativeF(x) of the functionf(x) = 2 sinxthat satisfiesF() = 2.
8. Find the functionf(x) withf(x) =2 for allx,f(0) = 42,f(0) = 1 andf(0) = 5.
9. A ball is thrown upwards with a speed of 48 ft/sfrom the roof of a 350 feet building. Assuming that near the surface of the earth the acceleration due to gravity is 32ft/s2, answer the following questions:
10. (a)Find the law of motion for the ball.
11. (b)When the ball reaches the highest point?
12. (c)When the ball hits the ground?
13. (d)What is the total distance that the ball has covered from the moment it was thrown until it hit the ground?

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