Spring 2014--CALCULUS 101A -- Test #1 (Ch 1) ... f . Find the following

Spring 2014--CALCULUS 101A -- Test #1 (Ch 1) ... f . Find the following

Spring 2014--CALCULUS 101A -- Test #1 (Ch 1)
NAME:-----------------------------------
Total: 100 points
1. For the function f (x) = x 2 + 6 , where x ≥ 0 . Find the following
a)
Find the inverse of the function.
b)
Verify the relationships: f ( f −1 (x)) = x and f −1 ( f (x)) = x .
(12 pts.)
2. When the Celsius temperature is 15 degrees, the corresponding Fahrenheit temperature is 59
degrees. When the Celsius temperature is 110 degrees, the corresponding Fahrenheit
temperature is 230 degrees. Let C represent the Celsius temperature and F the Fahrenheit
temperature.
(10 pts.)
a) Express F as an exact linear function of C.
b) Solve the equation in part (a) for C.
c) For what temperature is F = C
3. Verify that the following are identities:
a)
cos 2 θ (1+ tan 2 θ ) = 1
cos(A − B)
b)
= tan A + cot B
cos Asin B
c)
sin 2A cos2A = sin 2A − 4sin 3 A cos A
(6 pts. ea.)
4. Give the equations of any vertical, horizontal asymptotes and find the X-Y intercepts
x2 − 5x + 4
of: f ( x) =
.
(10 pts.)
x2 − 4
5. Find the exact value of the following without using a calculator.
"− 3%
'=
a. sin −1 $$
'
# 2 &
"− 2 %
" −1 %
'
b. cos−1 $ ' + sin −1 $$
'
2
#2&
#
&
(5 pts. ea)
6. Solve for x: log4 ( x + 3) + log4 (2 − x) = 1
(6 pts.)
7. Solve: 4 x − 2 x +1 = 3
3
2
8. Solve: (2 x + 1) = 64
(6 pts.)
(6 pts.)
ln(1 + e2 x ) = 2 x + ln(1 + e−2 x ).
(6 pts.)
10. Solve the equation for exact solutions over the interval [0,2π).
8cos2 x − 6 = 0 .
(6pts.)
9. Show that
11. Find the amplitude, period, and the phase shift, then sketch one cycle of the graph:
π
y = −2sin(2x − )
2