POSSIBLE SOLUTION(S)

1. Plot the graph of f[x] = x^2 on [-10,10].

2. Plot the graph of g[x] = Cos[x] on [-2 Pi, 2 Pi].

3. Predict and sketch the shape of the graph of h[x]=x^2 + Cos[x] on [-10,10].

4. Plot the graph of h on [-10,10] to verify your conjecture. What do you observe?

5. Find all values of x for which h has relative maximum or minimum values. Explain your results.

6. Predict the shape of the graph of h7[x] = x^2 + 7 Cos[x] on [-10, 10].

Having seen that the graph may not oscillate, students should be more careful to ask themselves if the amplitude of the cosine curve is great enough to result in local minima. Careful students should offer a graph that looks quite close to the real thing.

7. Plot the graph of h7 on [-10,10] to verify your conjecture. What do you observe?

8. Find all values of x on [-10,10] for which h7 has relative maximum or minimum values.

9. Consider the function hA[x]=x^2 + A cos x on [-10,10] where A is a constant. For what values of A will hA have relative extreme values? How many relative extreme values can we expect? Justify your answer.