Homework Assignment 12 (Written)
Chapters 4 and 5
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1. (12 points) Six statements are written below. Determine whether each statement is true or false (no work need
be shown for this problem).
(2 points) The function tan−1 (x) has as its domain [0, π] and has as its range the interval [−1, 1].
(2 points) L’Hopital’s rule can be used to conclude lim
et = 0.
(2 points) If the line y = m(x − a) + b is tangent to the graph of y = ex , then m = b = ea .
(2 points) The function f (x) = xx satisfies f 0
(2 points) If the invertible, differentiable function g(x) passes through the point (2, 5) with tangent line y − 5 = 8 − 4x then
g −1 (x) passes through the point (5, 2) with tangent line y − 2 = (x − 5).
(2 points) The derivative of the function f (x) = 12log12 x is the constant function 1.
2. (12 points) An open top box is to be constructed from rectangular pieces of cardboard. The base will be a square
cut from sturdy and more expensive material, one that costs $4.00 per square foot. The four walls will be cut from more
affordable material that costs $1.00 per square foot. What are the dimensions of the least expensive box one can design that
will enclose 16 cubic feet of volume? How expensive will this box be to construct?
3. (12 points)
(a) (6 points) Compute the derivative of y = ln
√cosh x − 1 · (x4 + 2)
x · (2x + 1)
(b) (6 points) Compute the derivative of f (x) = (sinh x)
4. (12 points) The function
f (x) = A cosh (Cx) + B sinh (Cx)
features three constants A, B and C. Answer the following questions to determine the values of A, B and C (using the
assumption that C > 0) so that f (x) satisfies the conditions
f 00 (x) = 36 · f (x)
f (0) = −2, f 0 (0) = 42
(A) (7 points) Use condition (1) to determine the value of the constant C > 0.
(B) (5 points) Use conditions (2) to determine the values of A and B
Note: Condition (1) is an example of a differential equation, and condition (2) is often called “initial conditions.” Solving differential equations subject to initial conditions is a main topic of study in various other mathematics and applied mathematics
5. (12 points) Another example of a differential equation is the following: the population, P (t), of a certain species
native to Houston – Instructorus Mathematicae – changes depending on time, t. Specifically, the rate of change in population
is proportional to the population level with constant of proportionality given by 3. In other words,
P 0 (t) = 3P (t).
(A) (5 points) Show that the function P (t) = 8e3t satisfies the differential equation above with initial condition P (0) = 8.
(B) (2 points) What is the initial rate of change in the population level of Instructorus Mathematicae (i.e. what is the rate
of change when t = 0?)
(C) (3 points) What is the rate of change in the population level of Instructorus Mathematicae at the point in time when
the population is P = 33?
6. (12 points) Evaluate the following limits.
ln 1 + x1
(A) (4 points) lim
arctan(x) − π/2
(B) (4 points) lim sinh x · csc x
(C) (4 points) lim
cos x + sin x
7. (7 points) Lets compute a derivative in reverse! Given the function f (x) = ex + sinh x +
function F (x) that satisfies F 0 (x) = f (x) and F (0) = 3. (Be sure to explain why F 0 (x) = f (x).)
8. (7 points) Find the equation of the line tangent to the graph of y = arctan (sinh(x)) at x = 0.
9. (7 points) The function f (x) = sinh(x) is one-to-one and so is invertible. Using our rule / formula for relating
the derivative of an inverse to the derivative of the original function, find the slope of the line tangent to the graph of
y = f −1 (x) = arcsinh(x) when x = 0.
10. (7 points) Several statements are provided below. Determine whether each one is true or false.
(2 points) If the function f (x) = ex + x2 + 1 is differentiated a total of 2022 times, the resulting function will be ex + 1.
(2 points) The largest product of two positive numbers that satisfy 2x + 3y = 5 is 25/24.
(2 points) Using logarithmic differentiation to compute the derivative of a function y(x) will, at some computational step,
involve the expression .
(1 point) I have studied hard and learned lots of Calculus in preparation for our upcoming Test 4.
(Hint: the answer is true.)
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