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Math 148.05A - Review for Final
(1) Chapter 10 - Limits and Continuity
(a) 10.1 Limits
(i) If limx→4 f (x) = 3 and limx→4 g(x) = 5, find limx→4 (4f (x) − 6g(x).
2
(ii) limx→3 x −2x−3
=
x−3
(2+h)3 −(2+h)−(23 −2)
h
1
limx→0 (1 + x) x =
x
limx→0 |x|
=
(iii) limh→0
=
(iv)
(v)
(b) 10.2 One-sided and infinite limits.
(i) limx→0+ x1 =
(ii) limx→0− x1 =
(iii) limx→∞ ln=x
3
2 +3x−12
(iv) limx→−∞ 4x3x−x
=
3 +10x−4
2
x +4
(v) limx→∞ −2x
3 +x =
(vi) limx→−1+ ln(x + 1) =
(c) 10.3 Continuity.
(i) What is the definition of continuity? I.e. A function f (x) is continuous at x = a
when...?
(ii) What does continuity on an interval (a, b) look like in terms of the graph of a
function?
(iii) What kinds of functions are continuous?
(iv) Is the following function continuous? Why or why not? If not, where does it
have discontinuities?
(
x2 ,
x≤1
f (x) =
x + 1, x > 1.
(v) Where is the following function continuous?
g(x) =
(x + 4)(x − 2)
.
(x + 2)(x − 3)
(vi) If f (x) and g(x) are continuous for all real numbers, where are the following
functions continuous?
(f + g)(x),
(f − g)(x),
(f g)(x),
( fg )(x),
(f ◦ g)(x).
(2) Chapter 11 - Differentiation.
(a) 11.1 Idea of derivative, definition.
(i) Let f (x) = x2 + 2x + 3. Write a formula for the slope of the secant line from
x = 2 to x = 2 + h. (Your answer should depend on h.)
(ii) Given your answer to the previous problem, write an expression (involving a
limit) for the slope of the tangent line to f (x) at x = 2. Evaluate your expression
to get a number.
(iii) Suppose a function g(x) is differentiable on the interval (10, 100). Is g(x) continuous on the interval (10, 100)?
(iv) Suppose a function h(x) is continuous on the interval (−1, 1). Is h(x) differentiable on the interval (−1, 1)?
(v) What does a function that is differentiable look like, in terms of its graph?
(vi) What does a non-differentiable function look like, in terms of its graph? Give
examples.
(b) 11.2 Derivative rules.
(i) Find the derivative of x3 using the definition of the derivative and using the
power rule. Check that they agree.
(ii) If f 0 (2) = 4 and g 0 (2) = −1, what is (4f − 2g)0 (2)?
(iii) Find the equation of the tangent line to y = x3 − x + 1 at the point (2, 7).
(iv) Find the derivative of f (x) = (x2 − 2)(x − 1 − x1 .
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(c) 11.3 Derivative as rate of change.
(i) If p(t) gives you the position of an object at time t, what does p0 (t) represent?
(ii) If marginal revenue for q = 10 is −5, estimate how much revenue changes if
quantity changes by 0.75.
(iii) Given a function f (x), describe the relationship between the average rate of
change of f over the interval (1, 1 + h) and the instantaneous rate of change of
f at x = 1.
√
(iv) A manufacturer models their costs using the function c(q) = 41 q 2 −2q+ q. What
is their average cost when q = 5? What is their marginal cost when q = 5?
(d) 11.4 Product rule and quotient rule.
(i) f (4) = 2, f 0 (4) = 5, g(4) = −1 and g 0 (4) = 12 . What are (f g)0 (4) and (f /g)0 (4)?
(ii) Find the derivative of (x3 − 2x2 + 3x − 4)ex .
(iii) If demand for your product is q, and the price of your product is p(q), then your
revenue is r(q) = qp(q). Find a formula for marginal revenue. (Your answer
should involve p.)
√
√
3x
(iv) Find the derivative of xx+
2 +1 .
(e) 11.5 Chain rule.
(i) Write the chain rule in two different ways.
(ii) If demand for your product at a time t is given by the function q(t), and the
price for your product is given by p(q), find a formula for revenue as a function
of time r(t). Find a formula for
√ marginal revenue as a function of time.
(iii) Find the derivative of f (x) = x−1 + x ln |x−2 + 2|.
(iv) Suppose selling q units brings in r(q) = q ln(q), and m employees can produce
q(m) = m + 10
m units. If you have 20 employees, estimate how much revenue
dr
changes if you hire one more employee. (Recall that dm
is called the marginal
revenue product.)
(3) Chapter 12 - Topics in Differentiation
(a) 12.1 Derivatives of logarithms.
(i) Draw a graph of ln |x| and its derivative on the same set of coordinate axes.
(ii) Use the change of base formula for logarithms to find a derivative rule for loga x
for any 0 < a, a 6= 1.
(iii) Find the derivative of log2 (x3/2 + x1/2 ).
(iv) Find the derivative of ln(x2 ex ln x).
(v) Find the equation of the tangent line to y = x + ln |x + 1| at x = − 12 .
(b) 12.2 Derivatives of exponentials.
(i) Sketch the graph of ex and the tangent lines at x = −2, 0, 2. Compare the slopes
to the value of ex at those points.
(ii) Use the chain rule and the derivative of ex to find a derivative rule for ax for
a > 0. Does your rule work for a = 1?
(iii) The balance of an account that earns continuously compounding interest at a
nominal rate of 5% per year is given by P (t) = P0 e0.05t dollars, with t in years,
where P0 is the initial balance. What is the rate of change of the ballance when
t = 4? What is the proportional rate of change of the balance when t = 1, 2, 3, 4?
2
1/2
(iv) Find the derivative of ex ex 2x .
(c) 12.3 Elasticity of demand.
(i) What is the definition of elasticity of demand?
(ii) What does it mean for demand to be elastic and inelastic? Give examples of
products likely to have elastic and inelastic demand.
(iii) If the price function for a product is p(q) = 21 q + 2 ln(q), what is the point
elasticity of demand when q = 5?
1 2
(iv) If the demand function for a product is q(p) = 10
q − q + 20, what is the point
elasticity of demand when p = 2?
(v) If point elasticity of demand is η = 2 and current demand for a product is 20,
dr
what is dp
for the current price? If you decreased your price by 50 cents, how
would that affect revenue?
Math 148.
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(d) 12.4 Implicit differentiation.
√
(i) If ln y + y = x, find y 0 .
√
(ii) Find the equation of the tangent line to 12 y 2 + x2 = 2 at the point (1, − 2)
(iii) If y + ey = x − ln x, find y 0 .
(e) 12.6 Newton’s Method
(i) for f (x) = x3 +2x, draw a diagram showing the first three iterations of Newton’s
Method, with x0 = 5.
(ii) What are a couple ways Newton’s Method might fail? Give examples.
(iii) Currently, demand for a product your company produces is q = 15. You know
that marginal revenue is M R(15) = 3, and you estimate that M R0 (15) = −2.
Use Newton’s Method to come up with a price for your product that will increase
revenue. (Hint: recall that revenue is maximized when M R(q) = 0.)
(iv) Starting with x0 = 2, estimate a root of the function f (x) = −ex + x + 2 to 5
degrees of accuracy.
(f) 12.7 Higher order derivatives.
d2 p
(i) If p(t) gives the position of an object at time t, what is dp
dt ? What is dt2 ?
(ii) How many nonzero derivatives does ex have? How many nonzero derivatives
does −3x3 + 2x2 − x + 1 have?
(iii) If y = x + x2 ln |x|, find y 000 .
(iv) If cost is given by c(q) = 61 x2 − x + 2, how fast is marginal revenue changing
when q = 8?
(4) Chapter 14 - Integration
(a) You need to measure the size of a circular oil slick. A surveying crew tells you that
the diameter of the slick is 550 feet, plus or minus 5 feet. Find the area of the slick,
and estimate the error in the value you calculate.
(b) If y = 2u + ln u and u = x2 + x1 , find dy in terms of dx.
(c) Estimate e0.02 using differentials.
(d) What is the relationship between an indefinite integral of a function f (x) and an
antiderivative of f (x)?
(e) Given a particular
function f (x), how many indefinite integrals does it have?
R
(f) What is f 0 (x) dx?
R
(g) If v(t)R is the velocity of an object at time t, what is v(t) dt?
(h) Find e√5x dx.
R 1+√x
√
(i) Find
dx.
R 1 x 2x
(j) Find 0 (x2 +1) ln(x2 +1) dx.
R3
(k) Draw a diagram of the area given by the integral 1 x1 dx.
(l) What is the fundamental theorem of calculus? Why is it important?
2
(m) If marginal revenue is M R(q) = q 2 + q+1
, what is the revenue function r(q)?
(n) Find the area between the curves x = 1 − y 2 , y = x − 1 and y = 2.
1
(o) If marginal cost is (q+1)
2 , estimate the change in costs if you change production from
q = 5 to q = 6. Now find the exact change in costs if you make that change in
production.
(p) A falling object is acted upon only by the force of gravity, so its acceleration is given
by a(t) = −9.8 m/s2 . After 5 seconds, the object hits the ground (i.e. height 0)
moving at a velocity of −40 m/s (the minus sign indicates that the object is moving
downwards). Find a formula for h(t), the height of the object at time t. What is the
initial velocity of the object?