Semester Exam Review Too
1.
2.
3.
Determine if the curve is a function:
a.
Determine if the curve has an inverse :
a.
f(x) = x2 + 2x
Name: _______________________________
b.
f(x) = -x3 +x2-2
b.
Graph the function, indicating all extrema (intercepts, asymptotes, local min/max).
a. f(x) = -x3 + 5x + 2
b. f(x) =
𝒙+𝟏
𝒙𝟐 −𝒙−𝟐
4.
Find the domain of f(x) = √𝒙𝟐 + 1
5.
Find the inverse of f(x) = x2+ 4
6.
Determine the number of solutions (and tell what they are) for: √𝒙𝟐 + 𝟐 = x3 – 1
7.
Convert to exponential form :
a.
8.
𝟐𝒙
√𝒙
Convert to root/fractional form :
a. 15-1/4
b.
𝟓𝒙
𝟏𝟎𝒙𝟐
b. -3x-5
𝟏
9.
Find g(f(x)) for f(x) = x2 and g(x) = 𝒙𝟐−𝟏
10.
Approximate the value of: 𝟑 𝒍𝒐𝒈𝟖 𝟔 (to the hundredths place).
11.
Rewrite as a single log: ln 9 – 3 ln 3.
12.
Solve the equation for x :
a. 6 ln 2x = 12
b. 𝒆𝒍𝒏 𝟓𝒙 = 𝟏𝟏
13.
Determine if the lines are parallel, perpendicular, or neither.
F(x) = -2(x + 3) – 4
and
g(x) = -2x + 5
14.
Find all zeros of: f(x) = 2x3 – x2 + 9
15.
Identify functions f(x) and g(x) such that (f o g)(x) equals the given function: √𝒄𝒐𝒔 𝟑𝒙 + 𝟏
16.
Estimate the length of the curve using n = 4 intervals for y = (x – 2)2 in the interval [0, 8].
17.
Identify the limits from the graph of f(x).
a. 𝐥𝐢𝐦− 𝒇(𝒙)
𝒙→𝟐
b. 𝐥𝐢𝐦 𝒇(𝒙)
𝒙→𝟎
c.
18.
𝐥𝐢𝐦 𝒇(𝒙)
𝒙→−∞
Sketch a graph of f(x) and then find the 𝐥𝐢𝐦 𝒇(𝒙).
𝒙→𝟏
f(x) =
19.
-2
if x ≤ -1
3
if -1 < x ≤ 1
x+2
if x > 1
𝒙𝟐 + 𝟑𝒙−𝟏
𝒙→−𝟒 𝟐𝒙+𝟐
The value of the 𝐥𝐢𝐦
is?
20.
Is the function in problem #19 discontinuous anywhere? If so, where? If not, explain.
21.
Identify any asymptotes (vertical or horizontal) for f(x). Tell what intervals f(x) is
𝒙+ 𝟐𝟎
continuous.
f(x) = 𝒙𝟐+ 𝟓𝒙
22. The function g(x) =
𝐱 𝟐 −𝟓𝐱+𝟔
is
𝐱+𝟑
discontinuous at x = 3. Is this discontinuity non-removable or
removable? Explain.
23.
Find the 𝐥𝐢𝐦
24.
Evaluate:
(𝒙−𝟑)(𝒙+𝟒)−(𝒙−𝟑)
𝒙→𝟐
(𝒙+𝟑)
by showing the algebraic work.
𝒙−𝟏𝟔
𝐥𝐢𝐦 𝟏𝟔−√𝟒𝒙.
𝒙→𝟒
25. On what interval is the function f(x) = √(𝐱 + 𝟒) + 8 continuous?
26. Evaluate the 𝐥𝐢𝐦
𝟐𝒙𝟑 + 𝟑𝒙−𝟔
𝒙→∞ 𝒙𝟑 + 𝟒𝒙𝟐 −𝟏
.
27. For the position function, f(t) =𝒕𝟐 + 𝒕, find the instantaneous velocity at a = 1 knowing the
instantaneous velocity function is 𝐥𝐢𝐦
𝒉→𝟎
28.
𝐟(𝐚+𝐡)−𝐟(𝐚)
.
𝒉
Sketch a tangent line at the given point for the graph.
a. x = 0
b. x = -4
c. x = 5
29.
List the points in order of increasing slope of the tangent line.
B
A
30.
Use the limit definition: 𝐥𝐢𝐦
𝒉→𝟎
𝐟(𝐚+𝐡)−𝐟(𝐚)
𝒉
C
, to find the indicated derivative.
𝑓′(5) 𝑓𝑜𝑟 𝑓(𝑥) = 𝑥 2 − 4𝑥
D
31.
Use the position function f(t) -8t2 + 2t + 4 to find the instantaneous velocity and
acceleration at time a = 5.
32.
Given the position function s(t )  16t 2  100t , give the velocity at t = 8 seconds and tell
whether or not the object is rising or falling at this time.
33.
Use the graph given below to sketch the corresponding graph of its derivative.
34.
Find a value c satisfying the Mean Value Theorem for f(x) = x4 + 4x + 1 on the interval [0, 2].
35.
Compute the derivative of:
a. f(x) = 5x3 + 2
b. f(x) = √4𝑥 + 2
c. f(x) = (x3 + 1)(x2-4x + 2)
5𝑥−3
d. f(x) = 4𝑥+1
e.
f.
g.
f ( x)  sin 4 x 2
f(x) = 5 tan x –csc 4x
f ( x)  ( x 2  6)7
h. f(x) = (x2-1)
𝑥 2 +3𝑥
𝑥 +2
i. h(x) = ln (x3 + 1)
36. Find the derivative of the function; be careful, it is NOT the first derivative you are finding.
37.
38.
a. f’’(x)
for
f(x) = xe3x
b. f”’(t)
for
f(t) = 𝟓𝒕𝟒 − 𝟏 + 𝒕𝟑
c. f(4)(x)
for
f(x) = x6 + 3x5 – 20
𝟓
Find an equation of the tangent line to y = f(x) at x = a.
a. f(x) = x3 – 2
a=0
b. f(x) = x cos x
a = π/2
Find a function for the given derivative.
a. f’(x) = 5x + 6
b. f’(x) = √𝒙
c. f’(x) = 5x4
d. f’(x) = 8x7 + 10x9 - 6