Review for Final
The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter
4. Below are the topics to study:
Chapter 2
• Find the exact answer to a limit question by using the limit rules
• Determine if a function is continuous (or discontinuous) at certain points, and
explain why
• Solve a limit problem involved with infinity
• Use the Squeeze Theorem to prove the answer to a limit question
• Use the Intermediate Value Theorem to prove a solution to an equation exists
• Estimate the slope of a tangent line (instantaneous rate of change) by drawing a
tangent line and estimating the slope
• Estimate the slope of a tangent line using the "forward-backward" technique
• Find the derivative by using the formula
• Sketch the graph of the derivative of a function
• Know at what points the derivative does not exist
Chapter 3
• Find the equation of a tangent line and/or normal line (of an explicit or implicit
equation)
• Find and simplify first and second derivatives (using the various rules)
• Use product and quotient rules (example: if f(3) = 4, f’(3) = 5, g(3) = -1 and g’(3)
= 2, what is the Derivative of f(3)g(3)?)
• Implicit differentiation
• Derivatives of logarithmic and inverse trig functions
• Logarithmic differentiation
• Linear approximation
• Using a differential to estimate errors
• Prove the following using the definition of derivatives: derivative of trig
functions.
• Prove the following: derivative of the natural logarithm, derivative of inverse trig
functions, power rule
• Taylor series
Chapter 4
• Related rates (similar to questions from section 4.1)
• Use derivatives to determine critical points, inflection points, maxima, minima,
direction and concavity (sections 4.2-4.3)
• Use L'Hopital's rule to determine limits
• Solve optimization problems
• Newton’s method
• Find the anti-derivative of a function
Sample Questions
1) Solve the following limit problems, using the method indicated:
t2 − 9
(use algebraic techniques, as in chapter 2)
a) xl→i−3m 2
2t + 7t + 3
1 − x + l nx
b) lx →i 1m
1 + c o πsx
(use L’Hospital’s Rule)
t2 − 9
(use any method)
c) lx →i ∞m 2
2t + 7t + 3
2) For each function below, find the derivative dy/dx
a) y = 4e5x – x2 + 6
8x 3 − 2 x
b) y =
c) 3xy2 + 9x = 4xy
d) y =
tan( x)
x3 − 5
e) y = x
x
(hint: use logarithmic differentiation)
3) Consider the function f(x) = tan-1(x)
a) Find the linear approximation at the point a = 1.
b) Use the linear approximation from part (a) to estimate tan-1(1.1)
c) Find the 3rd order Taylor polynomial (let n = 3, a = 1)
d) Use the Taylor polynomial from part (c) to estimate tan-1(1.1)
e) the “exact” value of tan-1(1.1) is 0.832981267. Which answer, from part (b) or
(d), is closer?
4) Two straight roads intersect at right angles in Newtonville. Car A is on one road
moving toward the intersection at a speed of 60 miles per hour. Car B is on the other road
moving away from the intersection at a speed of 40 miles per hour. (See the diagram
below). When Car A is 2 miles from the intersection and Car B is 4 miles from the
intersection, how fast is the distance between the cars changing? Is the distance
increasing or decreasing?
5) Find the absolute minimum and maximum values of y = x3 – 9x + 8 on the interval
[-3,1].
6) On what intervals is f(x) = e − x concave up?
2
7) A function has a second derivative f″(x) = 5x2 + 4x. Furthermore, f ′(0) = 2 and f(1) =
6. Find the function.
8) Find the dimensions of a rectangle of largest area that can be inscribed in an equilateral
triangle of side L = 12 cm, if one side of the rectangle lies on the base of the triangle.
9) Use the definition of the derivative to prove that if y = sin x, then y’ = cos x
10) Use logarithmic differentiation to prove the power rule.
11) Prove that if y = sin (x), then y ′ =
-1
1
1− x2
. (hint: re-write as x = sin y)
12) Find the antiderivative F(x) of the following:
a) f(x) = 4x2 – 2x + 6
b) f(x) = cos(x)
1
c) f(x) =
x
+ x3/ 4