Calculus Answers

) Consider f(x)=ax+xln(bx) for a>0, b>1, and x>1.

Find f′(x): f′(x)=

Based on your expression for f′(x), is f(x) increasing or decreasing? (Enter increasing or decreasing.)

(Be sure that you can see why this is true for all values x>1.)

Find f′′(x): f′′(x)=

Based on your expression for f′′(x), is f(x) concave up or concave down? (Enter up or down.)

(Be sure that you can see why this is true for all values x>1.)

For what values of x is the graph of y=xe^−2x concave down?

Trace the curve y =
\sqrt[3]{ {x}^{2} - 1}
3

x
2
−1
​

and state all the properties you use to trace it

f¹₃ (⨯)=cotx4.siny²x³

If k ≥ 1, the graphs of y = sinx and y = keex
intersect for
x ≥ 0. Find the smallest value of k for which the graphs are tan￾gent.
k =
What are the coordinates of the point of tangency?
x = , y = .

The area of a square with side s is A(s) = s
2
.
What is the rate of change of the area of a square with respect
to its side length when s = 3

At a time t seconds after it is thrown up in the air, a tomato is
at a height (in meters) of f(t) = =4.9t
2 +50t +5 m.
A. What is the average velocity of the tomato during the
first 4 seconds? (Include units.)
B. Find (exactly) the instantaneous velocity of the tomato
at t = 4. (Include units.)
C. What is the acceleration at t = 4? (Include units.)
D. How high does the tomato go? (Include units.)
E. How long is the tomato in the air?

At a time t seconds after it is thrown up in the air, a tomato is
at a height (in meters) of f(t) = =4.9t
2 +50t +5 m.
A. What is the average velocity of the tomato during the
first 4 seconds? (Include units.)
B. Find (exactly) the instantaneous velocity of the tomato
at t = 4. (Include units.)
C. What is the acceleration at t = 4? (Include units.)
D. How high does the tomato go? (Include units.)
E. How long is the tomato in the air?

A spherical snowball is melting in such a way
that its diameter is decreasing at rate of 0.1 cm/min. At what
rate is the volume of the snowball decreasing when the diameter
is 13 cm. (Note the answer is a positive number)

A voltage V across a resistance R generates a current I =
V/R. If a constant voltage of 22 volts is put across a resistance
that is increasing at a rate of 0.2 ohms per second when the re￾sistance is 5 ohms, at what rate is the current changing?