# CLEAR Calculus

## Lab 10: Taylor series - part 1

Function 1

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$f(x) = \frac{1}{1 + x}$$ centered at $$a = 2$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$f$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$f$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$f(x) = \frac{1}{1 + x}$$ centered at $$a = 2$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph so that

1.  The interval of convergence fills the entire width of the page

2.  The graphs of the six Taylor polynomials cover most (but not quite all) of the vertical space on the page.

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$f$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(-0.6)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.

Function 2

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$g(x) = \ln(x)$$ centered at $$a = 2$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$g$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$g$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$g(x) = \ln(x)$$ centered at $$a = 2$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph so that

1.  The interval of convergence fills the entire width of the page

2.  The graphs of the six Taylor polynomials cover most (but not quite all) of the vertical space on the page.

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$g$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(0.1)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.

Function 3

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$h(x) = e^{-x})$$ centered at $$a = -3$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$h$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$h$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$h(x) = e^{-x}$$ centered at $$a = -3$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph with $$-6 < x < 6$$ and $$-100 < y < 200$$.

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$h$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(-5)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.

Function 4

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$r(x) = 3\sin\left(\frac{\pi x}{5} \right)$$ centered at $$a = 5$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$r$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$r$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$r(x) = 3\sin\left(\frac{\pi x}{5} \right)$$ centered at $$a = 5$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph with $$0 < x < 20$$ and $$-10 < y < 10$$.

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$r$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(19)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.

Function 5

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$q(x) = (1 + 2x)^{-2}$$ centered at $$a = 0$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$q$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$q$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$q(x) = (1 + 2x)^{-2}$$ centered at $$a = 0$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph so that

1.  The interval of convergence fills the entire width of the page

2.  The graphs of the six Taylor polynomials cover most (but not quite all) of the vertical space on the page.

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$q$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(-0.4)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.

Function 6

Lab Preparation: Answer the following questions individually and bring your write-up to class.

1.  Find the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 for $$s(x) = \cos(2x)$$ centered at $$a = \frac{\pi}{2}$$.

2.  Plot the Taylor polynomials of orders 0, 1, 2, 3, 4 and 5 together with $$s$$ and describe what you see. Use a scale that clearly illustrates the convergence of the series to $$s$$.

Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually

1.  Find a formula for the full Taylor series for $$s(x) = \cos(2x)$$ centered at $$a = \frac{\pi}{2}$$.

2.  Use the ratio test to compute the radius of convergence for your Taylor series. Speculate on why it diverges beyond this radius.

3. Carefully redraw your graph with $$-\pi < x < 4\pi$$ and $$-5 < y < 5$$

4. Plot points from the following sequences on your graph until they are within 0.1 of the exact values for the function $$r$$. Plot each sequence above the corresponding $$x$$-value as if on a vertical number line. Label each point with its $$n$$-value.
5. Pick two of the approximation points from the sequence $$P_n(\pi)$$ on your graph and illustrate the error graphically. What is an algebraic representation of these errors? What is an algebraic representation of the error for any Taylor series approximation?

6. Using your graph, explain in detail what it means for the Taylor series of a function to converge to that function.