Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has the same y-value and slope as ln(x + ) at x = 0) and has the same curvature as ln(x + ) at x = 0. (c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + ) at x = 0. (d) Can you see a pattern yet? Can you guess what the degree 4 polynomial would be that has the same y-value and st, 2nd, 3rd and 4th derivatives as ln(x + ) at x = 0? Can you write down the degree 5 polynomial? Degree 6? Degree 0? (e) Graph ln(x+) and the degree 5 polynomial on the same screen of your calculator, for x. How similar do the graphs look? What happens if you use the degree 0 polynomial instead of the degree 5 one? 69
CHAPTER 7. CHALLENGE PROBLEMS 70 Example 2. (a) Find the equation of the tangent line for sin(x) at x = 0. (b) Find the equation of the parabola that is tangent to sin(x) at x = 0 (i.e. the parabola has the same y-value and slope as sin(x) at x = 0) and has the same curvature as sin(x) at x = 0. (c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as sin(x) at x = 0. (d) Can you see a pattern yet? Can you guess what the degree 4 polynomial would be that has the same y-value and st, 2nd, 3rd and 4th derivatives as sin(x) at x = 0? Can you write down the degree 5 polynomial? Degree 7? Degree 9? (e) Graph sin(x) and the degree 5 polynomial on the same screen of your calculator, for π/2 x π/2. How similar do the graphs look? What happens if you use the degree 9 polynomial instead of the degree 5 one?
CHAPTER 7. CHALLENGE PROBLEMS 7 Example 3. You should now know that ln(x + )=x 2 x2 + 3 x3 4 x4 + 5 x5... sin(x)=x 3! x3 + 5! x5 7! x7 +... The quantities on the right are called Maclaurin polynomials or Maclaurin series (which are infinite polynomials). (a) Find the Maclaurin polynomial for cos(x). Write it in the same way as I wrote the formula above for sin(x). (b) Take the derivative: d x dx 3! x3 + 5! x5 7! x7 +... and simplify it a little. Does your answer look familiar?
CHAPTER 7. CHALLENGE PROBLEMS 72 Example 4. You should now know that ln(x + )=x 2 x2 + 3 x3 4 x4 + 5 x5... sin(x)=x 3! x3 + 5! x5 7! x7 +... cos(x)= 2! x2 + 4! x4 6! x6 +... You should also know that when you take the derivative of the Maclaurin polynomial for sine, you get the Maclaurin polynomial for cosine. That s really nice. (a) Suppose this is correct for all functions (i.e. taking the derivative of the Maclaurin polynomial for f (x) gives you the Maclaurin polynomial for f (x). Use this to find the Maclaurin polynomial for, fill it in x + x + = x + x2 x 3 + x 4 x 5 +... (b) What do you get if you substitute x 2 in place of x everywhere in the above equation? Write it below: and simplify the powers of x. + = ( )+ ( )2 ( ) 3 + ( ) 4 ( ) 5 +... (c) What do you get if you integrate both sides of the above equation? Can you solve for C?
CHAPTER 7. CHALLENGE PROBLEMS 73 Example 5. You should now know that sin(x)=x 3! x3 + 5! x5 7! x7 +... tan (x)=x 3 x3 + 5 x5 7 x7 +... (a) Using your calculator, but not the sin button, try calculating sin(0.5) by plugging in x = 0.5 to the right hand side of the above formula. See how many terms you have to add before you get an accurate value. (You can use your sin button to double check your answer.) (b) What is tan ()? Using your calculator, but not the tan button, try calculating tan () by plugging in x = to the right hand side of the above formula. See how many terms you have to add before you get an accurate value. How can this help in calculating the value of π?
CHAPTER 7. CHALLENGE PROBLEMS 74 Example 6. You have now practiced starting with a function f (x) that you know. Then you have found a polynomial p(x)=a 0 + a x + a 2 x 2 + + a n x n such that p(x) has the same y-value, same slope, same second derivative, same third derivative,..., same nth derivative as f (x). You should have started to see patterns in every example. Can you give simple, pattern based formulas for the coefficients, that will work for any f (x)? (In other words your formula will be in terms of f.) a 0 = a = a 2 = a 3 = a 4 =... a n =
CHAPTER 7. CHALLENGE PROBLEMS 75 Example 7. Recap For any function f (x) that is differentiable at 0, you can find the Maclaurin polynomial p(x)=a 0 + a x + a 2 x 2 +...a n x n such that p(x) has the same y-value, same slope, same second derivative, same third derivative,..., same nth derivative as f (x), and that the formula for the coefficients of p(x) is a n = n! f (n) (0) Some examples of Maclaurin polynomials: ln(x + )=x 2 x2 + 3 x3 4 x4 + 5 x5... sin(x)=x 3! x3 + 5! x5 7! x7 +... cos(x)= 2! x2 + 4! x4 6! x6 +... x + = x + x2 x 3 + x 4 x 5 +... x 2 + = x2 + x 4 x 6 + x 8 x 0 +... tan (x)=x 3 x3 + 5 x5 7 x7 +... The Maclaurin polynomial is how your calculator actually calculates values of things like sine, cosine, ln(x), tan (x), etc. Maclaurin polynomials let us calculate things like π: π = 4 4 3 + 4 5 4 7 + 4 9... (a) Find the Maclaurin polynomial for e x. (b) Use the Maclaurin polynomial for e x to calculate e. (c) Find the Maclaurin polynomial for x +. (d) Use the Maclaurin polynomial for x + to calculate 2. (e) Are there any other basic functions you d like to know?