Calculus Online: Lab 6 Solutions

Question 1 (2 marks)

Which of the following graphs is a plot of the difference f(x)-g(x)?

Since f and g have the same value and slope at 0, the difference must have a value and slope of 0 at x=0. Also, since, g is greater than f, the difference is nowhere positive. Choice 'd' is the only graph that satisfies these properties.
a) b) c) d) e)

Question 2 (2 marks)

Which of the following quadratic polynomials provides the best approximation to exp(x) at x=0 in the sense that exp(x)-p(x) is flattest at the origin?
The polynomial below that has the flattest difference is the one that has the same value, first derivative, and second derivative as exp(x) at x=0. This polynomial is 'c'.
a) x2 b) 1 + x + x2 c) 1 + x + x2/2 d) 1 - x - x2 e) 1 - x - x2/2

Question 3 (2 marks)

If p(x) is written as

\[ p(x)=c_0+c_1x+\cdot\cdot\cdot+c_nx^n=\sum_{m=0}^{n} c_m x^m \] ,

what is p''(0)?

The second derivative is given by

\[ p''(x)=2c_2 + 2\cdot 3 c_3 x + 3\cdot 4 c_4 x^2 ... \] ,
so the answer is 2c2.
a) c2 b) c1 c) 2c2 d) c1+c2 e) c0

Question 4 (4 marks)

In the diagram below you see on the left a graph of cos(x). In the boxes above the graph you can enter the coefficients of a polynomial, which will also be plotted on the left. On the right you see the difference of the two functions.

Use the formula given above for cm+2 to find the coefficients of the Taylor polynomial of degree 8 of cos(x) at 0. Enter the coefficients into the appropriate boxes below. In each box you can enter a number, like 1.3, or a fraction of two numbers, such as -1/4. (The box will turn red if you have entered in something it doesn't understand, like 4*3).

To solve this question, you just need to find a polynomial with higher and higher derivatives that match cos(x) at x=0. The derivatives of cos(x) at 0 are easy to find (1, 0, -1, 0 1, 0, -1, 0, ...) so the problemis one of simple algebra.

Notice when you are done that the plot on the right is very flat near the origin, just as we discussed at the beginning of the lab.