The functions in Exercises 11-28 are all one-to-one. For each function, a . Find an equation for f − 1 ( x ) , the inverse function . b . Verify that your equation is correct by showing that f ( f − 1 ( x ) ) − x and f − 1 ( f ( x ) ) − x . f ( x ) = 2 x
The functions in Exercises 11-28 are all one-to-one. For each function, a . Find an equation for f − 1 ( x ) , the inverse function . b . Verify that your equation is correct by showing that f ( f − 1 ( x ) ) − x and f − 1 ( f ( x ) ) − x . f ( x ) = 2 x
Solution Summary: The author explains how to calculate the inverse of a one-to-one function: f(x)=2x
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
Step 1
Using the diagram of a right triangle given below, the relation between x, y, and z is
z²
= x²+
+12
x
Step 2
We must find dz/dt. Differentiating both sides and simplifying gives us the following.
2z
dz
dt
dx
2x.
+2y
dt
dx
dy
dz
x
+y
dt
dt
dt
2z
dy
dt
×
dx
(x+y
dt
dy
dt
An elastic rope is attached to the ground at the positions shown in the picture. The rope is being pulled up along the dotted line. Assume the units are meters.
9
ground level
Assume that x is increasing at a rate of 3 meters/sec.
(a) Write as a function of x: 0=
(b) When x=10, the angle is changing at a rate of
rad/sec.
(c) Let L be the the left hand piece of rope and R the right hand piece of rope. When x=10, is the rate of change of L larger than the rate of change of R?
○ Yes
○ No
4.1 Basic Rules of Differentiation.
1. Find the derivative of each function. Write answers with positive exponents. Label your derivatives with
appropriate derivative notation.
a) y=8x-5x3 4
X
b)
y=-50 √x+11x
-5
c) p(x)=-10x²+6x3³
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