The blood pressure in the systemic arteries is greatest

Exercise. Consider the parametric equations are

x(t)=1-t^2, y(t)=t-t^3 for -infinity<t < +infinity.

Use Equation (1) to express dy / dx  as a function of t.
 

Interpret the result of your calculation. What is the slope of the curve when t = -1?

Exercise. Consider the parametric equations are

x(t)=1-t^2, y(t)=t-t^3 for -infinity<t < +infinity.

Use Equation (1) to express dy / dx  as a function of t.
 

Interpret the result of your calculation. What is the slope of the curve when t = -1?

Consider a function f(x). Suppose its Taylor series centred at a converges for |x−a|<2. Suppose −1n≤f(n)(x)<n−1 for all |x−a|<2and for all n≥1     

Use Taylor's Inequality to find the best possible upper bound for |R3(x)| on its interval of convergence.  

Consider a power series ∑n=0∞cn(x+1)^n. Some partial information is known.

1. Each cn is a positive constant for all n≥0
2. limn→∞cn4n=∞
3. The series is convergent if we substitute x=−6 into the series.

Determine the radius of convergence of the series.

One function of the adventitia of the esophagus is to:

Question 8 options:

4. In this question, we will prove a theorem from one of the videos: Theorem 1. If f, g are bounded, integrable functions on [a, b], then so is f + g and: Z b a (f(x) + g(x)) dx = Z b a f(x) dx + Z b a g(x) dx
 (a) Prove that f + g is bounded on [a, b]. 
(b) Prove that for any partition P, UP (f + g) ≤ UP (f) + UP (g) and LP (f) + LP (g) ≤ LP (f + g). 
(c) Prove that I b a (f) + I b a (g) ≤ I b a (f + g) ≤ I b a (f + g) ≤ I b a (f) + I b a (g). Hint: you may use the result of part (b) and the properties of the lower sums and upper sums. 
(d) Conclude that f +g is integrable on [a, b] and that Z b a (f(x)+g(x)) dx = Z b a f(x) dx+ Z b a g(x) dx.