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Calculus 1
Test #2 Review Sheet
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Section 2.1: Tangent Lines and Velocity
The slope of the tangent line is the limit of the slopes of the secant lines as the second point of the secant
line approaches the tangent point:
f  a  h  f  a 
f  a  h  f a 
mtan  lim
 lim
h 0
h 0
(a  h)  a
h
Velocity is the slope of the tangent line of the position function f(t) at any given time:
v  a   lim
h 0
f  a  h  f a 
h
Section 2.2: The Derivative
Definition 2.1: The Derivative at a Point
The derivative of the function f(x) at x = a is defined as
f  a  h  f a
,
f '  a   lim
h 0
h
provided the limit exists.
If the limit exists, we say that f is differentiable at x = a.
Alternative Definition of the Derivative at a Point
f b  f  a 
f '  a   lim
b a
ba
Definition 2.2: The Derivative Function
The derivative of the function f(x) is defined as
f  x  h  f  x
f '  x   lim
,
h 0
h
provided the limit exists.
The process of computing a derivative is called differentiation.
Notice that differentiation produces a function whose output is the derivative of the function f(x) at any
value of x.
Things to notice when sketching graphs of f(x) or f (x) given the other function.
 Extrema of a graph have a slope / derivative of zero.
 Increasing regions of the graph have a positive derivative.
 Decreasing regions of the graph have a negative derivative.
Alternative derivative notations:
If y = f(x), then we also write the derivative of f(x) as:
dy df
d
f   x   y 


f  x
dx dx dx
Theorem 2.1: Relationship Between Differentiability and Continuity.
Calculus 1
Test #2 Review Sheet
Page 2 of 6
If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
Section 2.3: Computation of Derivatives: The Power Rule
Theorem 3.1: Derivative of a constant function
d
c  0 for any constant c
dx
Theorem 3.2: Derivative of f(x) = x
d
x 1
dx
Theorem 3.3: (Power Rule) Derivative of a positive integer power function
d n
x  nx n 1 for any integer n > 0
dx
Theorem 3.4: (General Power Rule) Derivative of any power function
d r
x  rx r 1 for any real number r.
dx
Theorem 3.4: Derivatives of Linear Combinations of Functions
d
 f x   g x   d f x   d g x 
1. The derivative of a sum is the sum of the derivatives:
dx
dx
dx
2. The derivative of a difference is the difference of the derivatives:
d
 f x   g x   d f x   d g x 
dx
dx
dx
3. The derivative of a product with a constant is the product of the constant and the derivative:
d
cf x   c d f x 
dx
dx
Higher order derivatives: (derivative of a derivative):
df
First order derivative: y '  f '( x)   x 
dx
d2 f
Second order derivative: y ''  f ''( x)  2  x 
dx
d3 f
Third order derivative: y '''  f '''( x)  3  x 
dx
d4 f
Fourth order derivative: y (4)  f (4) ( x)  4  x 
dx
Calculus 1
Test #2 Review Sheet
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Physics terminology:
The first derivative of the position function is called instantaneous velocity.
The second derivative of the position function is called instantaneous acceleration.
The third derivative of the position function is called the “jerk”.
Section 2.4: Product and Quotient Rules
Theorem 4.1: The Product Rule:
d
 p  x  q  x    p '  x  q  x   p  x  q '  x  .
If p and q are differentiable at x, then
dx 
Theorem 4.2: The Quotient Rule:
If n and d are differentiable at x, and d(x)  0, then
d  n  x  n ' x d  x  n  x d ' x


2
dx  d  x  
 d  x 
Section 2.5: The Chain Rule
Theorem 5.1 (Chain Rule):
d
f  g  x   f '  g  x g '  x 
dx
Or:
f
g   x    f  g  x   g '  x 
Leibniz notation for the Chain Rule:
dy dy du

.
dx du dx
Theorem 5.2: Derivative of an Inverse Function
If f is differentiable at x and has an inverse function g  x   f 1  x  , then
If y = f(u), and u = g(x), then y = f(g(x)), and
g  x 
1
, provided f   g  x    0 .
f   g  x 
Section 2.6: Derivatives of Trigonometric Functions
d
sin x  cos x
dx
d
cot x   csc 2 x
dx
d
cos x   sin x
dx
d
sec x  sec x tan x
dx
d
tan x  sec 2 x
dx
d
csc x   csc x cot x
dx
Calculus 1
Test #2 Review Sheet
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Section 0.5: Exponential and Logarithmic Functions
e x  e x
sinh  x  
2
x
e  e x
cosh  x  
2
sinh  x 
tanh  x  
cosh  x 
cosh 2  x   sinh 2  x   1
Section 2.7: Derivatives of Exponential and Logarithmic Functions
d
Theorem 7.1: a x  a x ln a
dx
d x
d x
e  ex ,
e  e  x
Theorem 7.2:
dx
dx
d
1
Theorem 7.3: ln x  
dx
x
Section 2.8: Implicit Differentiation and Inverse Trigonometric Functions
To perform implicit differentiation:
1. Take the derivative of both sides of the equation.
2. Solve for y.
Theorem 8.1: Power Rule for Rational Exponents
d r
x  rx r 1 .
For any rational exponent r,
dx
Derivatives of the Inverse Trigonometric Functions
d
1
, for –1 < x < 1
sin 1 x 
dx
1 x2
d
1
, for –1 < x < 1
cos 1 x 
dx
1 x2
d
1
tan 1 x 
dx
1 x2
d
1
, for |x| > 1
sec 1 x 
dx
x x2 1
d
1
cot 1 x 
dx
1 x2
d
1
, for |x| > 1
csc 1 x 
dx
x x2 1
Calculus 1
Test #2 Review Sheet
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Section 2.9: The Mean Value Theorem
Theorem 9.1 (Rolle’s Theorem): (The “What goes up must come down” Theorem)
If f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there is
a number c  (a, b) such that f  (c) = 0.
Theorem 9.2 (Corollary #1 to Rolle’s Theorem):
If f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and f(x) = 0 has two
solutions in [a, b], then f  (x) = 0 has at least one solution in (a, b).
Theorem 9.3 (Corollary #2 to Rolle’s Theorem):
For any integer n > 0, if f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and
f(x) = 0 has n solutions in [a, b], then f ‘ (x) = 0 has at least n-1 solutions in (a, b).
Theorem 9.4 (Mean Value Theorem):
If f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), then there is a number c
 (a, b) such that
f (b)  f (a )
f ' (c ) 
ba
Picture:
Theorem 9.5 (Corollary #1 to the Mean Value Theorem):
If f  (x) = 0 for all x in some open interval I, then f (x) is constant on I.
Theorem 9.6 (Corollary #2 to the Mean Value Theorem):
If f  (x) = g  (x) for all x in some open interval I, then for some constant c,
f (x) = g(x) + c for all x  I.
Practice:
Find all functions that have a derivative equal to 3x2 + 1.
Find all functions that have a derivative equal to x4 + 3x.
Calculus 1
Test #2 Review Sheet
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