1.    Find each of the following limits:
    (a) 
    (b) 
    (c) 
    (d) 
    (e) 

2.    Discuss the continuity of the graph at x = -2, 0, and 1.

---

Solution: 1.
    (a)  = 3--2=5
    (b)  = (-1/2)³ = -1/8
    (c)  = 2/DNE = Does not exist
    (d)  = 1.5 + 0 = 1.5
    (e)  = f(g(3)) = f(1) = 2
2.    The graph is removably discontinuous at x = -2 (since a general limit exists there), nonremovably discontinuous at x = 1 (as there is no general limit there), and continuous at x = 0.

---

Calculus II (BC): Parametric Equations

Given the plane curve, C, defined by the following parametric equations:

    (a)    Graph C for t on the interval (0,3].
    (b)    Find the rectangular equation for C.
    (c)    Adjust the domain of (b) so that it matches the domain of C.

Difficulty Rating: 

Source: Calculus: Concepts and Contexts, James Stewart, p. 305.

---

Solution:     (a) 

    (b)    In order to convert this to rectangular form, we usually try to solve one of the equations for t. In this case, we notice that the inside of x's square root is a perfect square; thus, solve for t:

Before we plug t into the y equation, we note that the fraction can be simplified--then we substitute:     (c)    In order to match the graph above, we clearly need to restrict the domain of the rectangular function to x > 1.

---

Home

Problem

Fun Calculus Stuff

Kelley's Books

Superbowl of Calculus