Grab bag problems: Week 5
Topics: definition of derivative; tangent lines; higher derivatives; concavity; derivative of exponentials; product rule; chain rule.
Note: We have learned and will learn many rules for computing derivatives of functions. There
are lots of problems (which you should do!) in the book that will give you practice with applying
those rules. The purpose of this week’s Grab Bag is not to give you practice with applying the
derivative rules. Rather, the purpose of the Grab Bag is to complement the type of questions that
you can find in the book, not to replace them. So, be sure that you can also do the questions from
the book!
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Problem 1. Suppose f (x) = 25 − x.
1. Compute f 0 (x).
2. Find the tangent line to the graph when x = 16.
3. Write an equation for the tangent line to the graph y = f (x) at the point (a, f (a)). (So, your
answer should depend on a.)
4. Suppose the tangent line to the graph of y = f (x) at the point (a, f (a)) intersects the x-axis
at the point (60, 0). What is a?
Problem 2. Suppose f (x) and g(x) are two differentiable functions. Assume that the tangent line
to the graph y = f (x) when x = 4 has equation y = 7(x − 4) + 3 and the tangent line to the graph
of g(x) when x = 4 has equation y = 5(x − 4) − 13. What are the equations for the tangent line to
the graph of y = f (x) + g(x) when x = 4 and y = f (x)g(x) when x = 4?
Problem 3. Suppose g(x) is defined by
(
4x2 + 3 if x < −1
g(x) =
.
8|x| − 1 if x ≥ −1
g(−1 + h) − g(−1)
from the definition of the function g.
h
g(−1 + h) − g(−1)
from the definition of the function g.
2. Compute lim
−
h
h→0
3. Is g differentiable at x = −1? Why or why not?
1. Compute lim
h→0+
Problem 4. Suppose f is function that is concave down, and for which f 0 (x) is positive for all x.
Which of the following scenarios could truthfully describe the values of f and f 0 ?
1. f (1) = 12, f (2) = 9, f 0 (2) = −2, f (3) = 8;
2. f (1) = 10, f (2) = 14, f 0 (2) = 5, f (3) = 17;
3. f (1) = 1, f (2) = 5, f 0 (2) = 3.5, f (3) = 8;
4. f (1) = −3, f (2) = 3, f 0 (2) = 4.5, f (3) = 8.
Problem 5. Suppose f (x) = e−x , g(x) = 3x − 2, and set h(x) = f (x) − g(x). On your midterm,
you computed h(0) = 3 > 0 and h(1) = 1e − 1 < 0, so by that by the intermediate value theorem,
there is some c between 0 and 1 so that h(c) = 0.
1
1. Compute h0 (x).
2. Is h0 (x) always positive, always negative, or sometimes positive/sometimes negative?
3. On your midterm, you saw that there is no negative number x so that h(x) = 0. Using your
answer from part 2, explain why the number c is actually the only real number (positive or
negative) for which h(c) = 0.
Problem 6. Suppose g is differentiable, and is an even function. Remember that g being even
means that g(−x) = g(x) for all x.
1. If you know g 0 (3) = 8, can you determine g 0 (−3)?
2. How about g 0 (0)?
The following problem is more abstract than we would ask you on a midterm, but nonetheless
highlights a really important concept from Calculus.
Problem 7. If h(x) is any differentiable function, we know that the tangent line to the graph of
h at the point (c, h(c)) has equation y = h0 (c)(x − c) + h(c). Define a linear function
Lh,c (x) = h0 (c)(x − c) + h(c).
Thus the tangent line to y = h(x) at the point (c, h(c)) has equation y = Lh,c (x).
Now suppose f and g are two differentiable functions. We know that the tangent line to the graph
of f at the point (a, f (a)) has equation
y = f 0 (a)(x − a) + f (a) = Lf,a (x)
and the tangent line to the graph of g at (f (a), g(f (a))) has equation
y = g 0 (f (a))(x − f (a)) + g(f (a)) = Lg,f (a) (x).
1. Using the chain rule, write down a formula for the linear function Lh,a if h(x) = f (g(x)).
2. Compute the composition of the two linear functions Lg,f (a) ◦ Lf,a .
What do you notice?
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