For Exercises 35-44, an equation of a parabola x − h 2 = 4 p y − k or y − k 2 = 4 p x − h is given. a. Identify the vertex, value of p , focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) x + 2 2 = y − 7 4
For Exercises 35-44, an equation of a parabola x − h 2 = 4 p y − k or y − k 2 = 4 p x − h is given. a. Identify the vertex, value of p , focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) x + 2 2 = y − 7 4
Solution Summary: The author calculates the vertex, value of p, focus and focal diameter of the parabola.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Finding The Focus and Directrix of a Parabola - Conic Sections; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=KYgmOTLbuqE;License: Standard YouTube License, CC-BY