MATH 131:100 Exam 2 Review
17 June, 2014
The second Exam will have three sections: one of Definitions, one for graphical concepts,
and a section of workout problems. For this Review, I will just give a sample of the types
of problems that I could ask, but any of the material we have covered From Section 2.6 to
Section 3.9 is fair game.
Definitions You Should Know:
1) The limit definition of the derivative of f at the point a
f (x) − f (a)
x−a
2) Another limit definition of the derivative of f at the point a
lim
x→a
f (a + h) − f (a)
h
3) Supposing that f and g are differentiable functions. Then the Product Rule states
that
d
[f (x)g(x)] = f (x)g 0 (x) + f 0 (x)g(x)
dx
4) Supposing that f and g are differentiable functions. Then the Quotient Rule states
that
d f (x)
g(x)f 0 (x) − f (x)g 0 (x)
=
2
dx g(x)
(g(x))
5) Supposing that f and g are differentiable functions. Then the Chain Rule states that
lim
h→0
d
f (g(x)) = f 0 (g(x)) · g 0 (x)
dx
6) A function f is differentiable at a if f 0 (a) exists (that is, the limit defined in 1 or 2
exists).
7) A function f is differentiable on an interval (a,b) if f is differentiable at every point
in that interval.
8) The Linearization of a function f at a point a is the function
L(x) = f (a) + f 0 (a)(x − a)
Other Topics:
ˆKnow how to compute derivatives from the limit definitions.
ˆKnow how differentiability and continuity are related (see Section 2.7).
ˆKnow how to determine properties of f from the graphs of f 0 and f 00 .
ˆKnow how to determine properties of f from its graph (i.e. increasing, decreasing,
concavity, etc.)
ˆKnow how to draw graphs of functions with certain derivative and second derivative
properties (as in Section 2.8).
ˆKnow all differentiation rules from Section 3.1.
ˆKnow properties of logarithms and exponential functions, and how they relate to one
another (including cancellation equations)
ˆKnow Product, Quotient, and Chain Rules and how to compute derivatives using them.
ˆKnow Point-Slope Form equation for finding the equation of a line between two given
points.
ˆKnow derivatives of all basic functions.
ˆKnow derivatives of Trig functions (from section 3.3)
ˆKnow how to apply differentiation techniques to solve real world type problems (see
Section 3.8)
ˆKnow how to form the Linearization of a function f at a point a, and how to estimate
a function using this.
ˆKnow how to use Differentials to approximate error.
Example Problems:
Problem 1) The following table shows the population of the world every decade from
1900 to 2000.
Year
Population (millions)
1900
1650
1910
1750
1920
1860
1930
2070
1940
2300
1950
2560
1960
3040
1970
3710
1980
4450
a) What was the average rate of change of the population from 1940 to 1950? From 1950
to 1960?
b) Use your answer from part (a) to estimate P 0 (1950).
Problem 2) Differentiate
f (x) = tan esin x
Problem 3) Differentiate
f (x) =
(no need to simplify)
ln x + 3x4
1 + x2
1990
5280
2000
6080
Problem 4) Find the equation of the tangent line to the curve y = x4 + 3x2 − 2x + 10
at the point (1, 14).
Problem 5) Differentiate
f (x) = 10x sin x
Problem 6) Differentiate
f (x) = log7 (ex )
Problem 7) The edge of a cube is measured to be 30cm with a possible error of 0.1cm.
Use differentials to estimate the maximum possible error in calculating the volume of the
cube.
Problem 8)
a) Find the linearization of the function f (x) = x3/4 at the point a = 16.
b) Use your answer from part (a) to estimate the value of (16.1)3/4 .