CALCULUS FINAL EXAM REVIEW
QUESTIONS WITH COMPLETE
SOLUTIONS
Section 1.5 What is the equation for an exponential function? - ANSWER-P(t)= p0a^t
Section 1.5 What is the equation for an exponential function that is compounded per
period? - ANSWER-P(t)=p0(1+r/n)^(nt)
Section 1.5 What is the equation for an exponential function that is compounded
continuously? - ANSWER-P(t)=Pe^(rt)
Section 1.5 If you were to be given the continuous rate of a function, how can you find
the effective rate? - ANSWER-e^r-1
Section 1.5 If you were to be given the compounded per period rate of a function, how
can you find the effective rate? - ANSWER-(1+r/n)^(n)-1
Section 1.6 natural log rules: multiplying ln, dividing ln, ln raised to a power, ln with e, ln
(1), and x raised to the zero power. - ANSWER-multiplying ln: ln(AB)= ln(A)+ln(B)
dividing ln: ln(A/B)= ln(A)-ln(B)
ln raised to a power: ln(A^p)=p(ln(A))
ln with e: ALWAYS CANCELS OUT ln(e^x) or e^lnx
ln(1)= 0
x^0= 1
Section 1.7 What do we call it when you want to find the time that will give you twice the
initial amount you started with? - ANSWER-Doubling time
Section 1.7 What do we call it when you want to find the time that will give you half the
amount you started with? - ANSWER-Half-life
Section 1.7 How do we find the present value of an exponential function that is
compounded continuously? - ANSWER-PV=Be^-rt
Section 1.7 How do we find the present value of an exponential function that is
compounded "n" times per year? - ANSWER-PV=B(1+r/n)^(-nt)
Section 1.9 How do we know if a function is a power function? - ANSWER-When it can
be written as y=k*x^p.
Section 1.9 When given a power function, if the two things are directly proportional,
what would the equation be? - ANSWER-thing1=k*thing2
, Section 1.9 When given a power function, if the two things are inversely proportional,
what would the equation be? - ANSWER-thing1=k/thing2
Section 2.2 What is the instantaneous rate of change? - ANSWER-The instantaneous
rate of change is just the derivative.
Section 2.2 If we know that f(x) is increasing, what can we tell about f'(x)? - ANSWER-if
f(x) is increasing then f'(x) is positive.
Section 2.2 If we know that f(x) is decreasing, what can we tell about f'(x)? - ANSWER-if
f(x) is decreasing then f'(x) is negative.
Section 2.2 If we know that the slope of the function is zero, what can we tell about f'(x)?
- ANSWER-if f(x) is zero then f'(x) is also zero.
Section 2.2 If we know that f(x) is concave up, what can we tell about the first and
second derivative? - ANSWER-if f(x) is concave up then f'(x) is increasing and f"(x) is
positive.
Section 2.2 If we know that f(x) is concave down, what can we tell about the first and
second derivative? - ANSWER-if f(x) is concave down then f'(x) is decreasing and f"(x)
is negative.
Section 2.2 If we know that the first derivative is concave up, what can we tell about the
second derivative? - ANSWER-if f'(x) is concave up then f"(x) is increasing.
Section 2.2 If we know that the first derivative is concave down, what can we tell about
the second derivative? - ANSWER-if f'(x) is concave down then f"(x) is decreasing.
Section 2.3 What is the equation for local linear approximation? - ANSWER-f(a+h)=f(a)
+h(f'(a))
Section 2.5 How do we know when profit is maximized? - ANSWER-Profit is maximized
when the marginal revenue equals the marginal cost.
Section 2.5 What happens when marginal revenue exceeds marginal cost? - ANSWER-
You are underproducing.
Section 2.5 What happens when marginal revenue is less than marginal cost? -
ANSWER-You are overproducing.
Section 3.1-3.4 How do we find the derivative of these formulas?
constant rule: f(x)=k
power rule: f(x)=x^k
exponential rule: f(x)=k^x
QUESTIONS WITH COMPLETE
SOLUTIONS
Section 1.5 What is the equation for an exponential function? - ANSWER-P(t)= p0a^t
Section 1.5 What is the equation for an exponential function that is compounded per
period? - ANSWER-P(t)=p0(1+r/n)^(nt)
Section 1.5 What is the equation for an exponential function that is compounded
continuously? - ANSWER-P(t)=Pe^(rt)
Section 1.5 If you were to be given the continuous rate of a function, how can you find
the effective rate? - ANSWER-e^r-1
Section 1.5 If you were to be given the compounded per period rate of a function, how
can you find the effective rate? - ANSWER-(1+r/n)^(n)-1
Section 1.6 natural log rules: multiplying ln, dividing ln, ln raised to a power, ln with e, ln
(1), and x raised to the zero power. - ANSWER-multiplying ln: ln(AB)= ln(A)+ln(B)
dividing ln: ln(A/B)= ln(A)-ln(B)
ln raised to a power: ln(A^p)=p(ln(A))
ln with e: ALWAYS CANCELS OUT ln(e^x) or e^lnx
ln(1)= 0
x^0= 1
Section 1.7 What do we call it when you want to find the time that will give you twice the
initial amount you started with? - ANSWER-Doubling time
Section 1.7 What do we call it when you want to find the time that will give you half the
amount you started with? - ANSWER-Half-life
Section 1.7 How do we find the present value of an exponential function that is
compounded continuously? - ANSWER-PV=Be^-rt
Section 1.7 How do we find the present value of an exponential function that is
compounded "n" times per year? - ANSWER-PV=B(1+r/n)^(-nt)
Section 1.9 How do we know if a function is a power function? - ANSWER-When it can
be written as y=k*x^p.
Section 1.9 When given a power function, if the two things are directly proportional,
what would the equation be? - ANSWER-thing1=k*thing2
, Section 1.9 When given a power function, if the two things are inversely proportional,
what would the equation be? - ANSWER-thing1=k/thing2
Section 2.2 What is the instantaneous rate of change? - ANSWER-The instantaneous
rate of change is just the derivative.
Section 2.2 If we know that f(x) is increasing, what can we tell about f'(x)? - ANSWER-if
f(x) is increasing then f'(x) is positive.
Section 2.2 If we know that f(x) is decreasing, what can we tell about f'(x)? - ANSWER-if
f(x) is decreasing then f'(x) is negative.
Section 2.2 If we know that the slope of the function is zero, what can we tell about f'(x)?
- ANSWER-if f(x) is zero then f'(x) is also zero.
Section 2.2 If we know that f(x) is concave up, what can we tell about the first and
second derivative? - ANSWER-if f(x) is concave up then f'(x) is increasing and f"(x) is
positive.
Section 2.2 If we know that f(x) is concave down, what can we tell about the first and
second derivative? - ANSWER-if f(x) is concave down then f'(x) is decreasing and f"(x)
is negative.
Section 2.2 If we know that the first derivative is concave up, what can we tell about the
second derivative? - ANSWER-if f'(x) is concave up then f"(x) is increasing.
Section 2.2 If we know that the first derivative is concave down, what can we tell about
the second derivative? - ANSWER-if f'(x) is concave down then f"(x) is decreasing.
Section 2.3 What is the equation for local linear approximation? - ANSWER-f(a+h)=f(a)
+h(f'(a))
Section 2.5 How do we know when profit is maximized? - ANSWER-Profit is maximized
when the marginal revenue equals the marginal cost.
Section 2.5 What happens when marginal revenue exceeds marginal cost? - ANSWER-
You are underproducing.
Section 2.5 What happens when marginal revenue is less than marginal cost? -
ANSWER-You are overproducing.
Section 3.1-3.4 How do we find the derivative of these formulas?
constant rule: f(x)=k
power rule: f(x)=x^k
exponential rule: f(x)=k^x
