Homework # 3 Due: 6/6/06 1. Find an equation of the tangent line to y = 2x + 1 at the point (4,3).
2. Draw a graph of a function f that has the properties g(0) = 0 g (0) = 3 g (1) = 0 g (2) = 1 3. Use the limit definition of the derivative to compu
Homework # 1 Due: 05/23/06 #1 Find the domain of the following functions 3x2 - 2x + 1 a) f (x) = 2 x - 4x - 21 3 b) f (x) = x + 3 - x - 2 + x2 - 1 1 c) f (x) = x2 - 1 #2 Graph the following piecewise function f (x) = #3 1.2 # 2 #4 1.2 # 12 #5 f
Quiz # 1 Name: 1. Find f (2 + h), f (x + h), and f (x + h) - f (x) if f (x) = x - x2 . h
2. Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, or transcendental function (
Homework # 3 Due: 6/6/06 1. Find an equation of the tangent line to y = 2x + 1 at the point (4,3).
The equation of a tangent line to y = f (x) at (a, b) is (y - b) = f (a)(x - a). First, we 1 write the equation as y = (2x + 1) 2
1 d (2x + 1) 2 dx
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Tangents and Velocities
Recall finding tangents numerically.
1.1
Tangents
Remember formula for slope of secant line. Definition 1. The tangent line to the curve y = f (x) at the point (a, f (a) is the line through the point with slope f (x) -
1
Maximum and Minimum Values
This is all about optimization problems. Definition 1. A function f has an absolute maximum at c if f (c) f (x) for x in the domain of f . f (c) is called the maximum value. A function has an absolute minimum at c if f
Homework # 6 Due: Never
1
New Material
1. Use the guidelines of this section to sketch the curve. (a) f (x) = 20x3 - 3x4 (b) f (x) =
x2 x2 -9
(c) f (x) = sin x 2. If 1200 cm2 is available to make a box with a square base and an open top, find the
Homework # 6 Due: Never
1
New Material
1. Use the guidelines of this section to sketch the curve. (a) f (x) = 20x3 - 3x4 i. Domain f is a polynomial, so its domain is all real numbers. ii. Intercepts The y-intercept is f (0) = 0. The x-intercepts a
Homework # 4 Due: 6/13/06 1. Differentiate the following: (a) f (x) = x2 (cos x)(sin x) (b) f (x) = (c) f () = (d) f (x) =
tan x-1 sec x sin (+tan ) 1+sec (x-1)4 (x2 +2x)5
(e) f (x) = sin tan
sin x
2. Find dy/dx by implicit differentiation: (a)
Homework # 2 Due: 5/30/06 1. Use the graph to find the following limits: lim f (x) lim f (x) lim f (x) lim f (x) lim f (x)
x2+ x0
x2-
x2
x-1
2. Sketch the graph of a function that satisfies all of the given conditions: lim f (x) = 1 lim f (x) =
Homework # 1 Due: 05/23/06 #1 Find the domain of the following functions 3x2 - 2x + 1 a) f (x) = 2 x - 4x - 21 The domains of 3x2 - 2x + 1 and x2 - 4x - 21 are all real numbers. So the domain of f is all real numbers except where x2 - 4x - 21 = 0. So
Homework # 2 Due: 5/30/06 1. Use the graph to find the following limits: lim f (x) lim f (x) lim f (x) lim f (x) lim f (x)
x2+ x0
x2-
x2
x-1
2. Sketch the graph of a function that satisfies all of the given conditions: lim f (x) = 1 lim f (x) =
1
Related Rates
So far we've been looking at the rate of change of one "thing". Maybe it's a particle or just a function or what have you. A lot of the time, however, when something changes it causes something else to change. For instance, say you'
1
Formulas
Remember, mathematics is all about being lazy. Using the limit definition of the derivative will get very difficult if we have to do it every time. Fortunately, we have methods for computing common derivatives like we have methods for co
1
1.1
Continuity
Definitions
Last time, I went over the direct substitution property. That says that for polynomials and rational functions, as long as a is in the domain, lim f (x) = f (a). It turns out that this property works for a lot of differ
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Tangents and velocities
On the first day, I talked very briefly about where calculus comes from. One branch comes from studying tangent lines to curves. Here's a more in-depth overview. A tangent line to a circle is basic. (draw picture) But a ta
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1.1
Trigonometry
Angles
I mentioned on Tuesday that we don't use degrees in calculus. Instead we use a unit called radians. Definition 1. One radian is the angle that gives an arc length equal to the radius. Since the circumference of a circle is
1
1.1
Functions
What is a function?
All a function is, is something that takes a number and turns it into another number. Example 1.1. Remember from geometry class the formula for a circle, A = r2 . This is a nice example of a function. It takes a