Math Derivatives Assignment - Calculus Problems & Solutions

Math 201-AS1-EN Assignment 5
Derivatives Fall 2022
•Prepare this assignment as you would any other, but do not
upload your work to L´ea. There will instead be a 15-minute
quiz at the beginning of class on Tuesday, October 4 (a Mon-
day schedule). The quiz will consist of problems from this
assignment—either identical to the assignment or with only
minor changes. You can use the answers with selected solu-
tions (posted on L´ea) to help you prepare, but you will have
to put away your notes before writing the quiz.
•Presentation counts! Derivatives are a special kind of limit,
so remember to write limh→0or limx→aat each step except
the last, where the limit is finally worked out by direct sub-
stitution. For this assignment, use only the definition of the
derivative, not the various differentiation rules we are about
to discuss in class.
•You are free to use the definition of the derivative in either
of the forms
f′(a) = lim
h→0
f(a+h)−f(a)
hor f′(a) = lim
x→a
f(x)−f(a)
x−a
However, if you require the derivative not only at a single
value of abut as a function of x(or whatever variable is
specified for f), then remember to replace aby xin your
answer. Alternatively, use either of the above forms with a
replaced by x:
f′(x) = lim
h→0
f(x+h)−f(x)
hor f′(x) = lim
t→x
f(t)−f(x)
t−x
(Notice that replacing aby xin the second form requires
that we also replace xby some other variable—in this case t.
If you worry that you might mix up the variables, consider
using only the first form.)
Problems
1. Find equations of both the tangent and normal lines to
the curve
y=x+ 2
x−3
at the point (2,−4).
2. (a) Find the slope of the tangent to the curve y= 2√x
at the point where x=a.
(b) Find equations of the tangent lines at the points
(1,2) and (9,6).
3. The displacement (in metres) of a particle moving in a
straight line is given by the equation of motion s= 1/t2,
where tis measured in seconds. Find the velocity of the
particle at times t=a,t= 1, t= 2, and t= 3.
4. If the tangent line to y=f(x) at (4,3) passes through
the point (0,2), find f(4) and f′(4).
(5–6)Each limit represents the derivative of some function f
at some number a. State such an fand ain each case.
5. lim
h→0
√9 + h−3
h6. lim
x→2
x6−64
x−2
(7–10)Trace or copy the graph of the given function f. Then
sketch the graph of f′below it. (Estimate the value of the
derivative at any value of xby drawing the tangent at the
point (x, f(x)) and estimating its slope. Assume that the
axes have equal scales.)
7.
x
y8.
x
y
9.
x
y10.
x
y
(11–15)Find the derivative of the function using the defini-
tion of derivative. State the domain of the function and the
domain of its derivative.
11. f(t) = 2t2−5t+ 3 12. f(x) = x
1−4x
13. f(x) = 1
x2−414. f(x) = 1
√1 + x
15. f(x) = √x2+ 4
16. The graph of fis given. State, with reasons, the num-
bers at which fis not differentiable.
x
y
−2 1

17. The figure shows the graphs of three functions. One is
the position function of a car, one is the velocity of the
car, and one is its acceleration. Identify each curve, and
explain your choices.
x
ya
bc
Hints and suggestions
•Hints for Problems 7–10: Any horizontal tangent on the
graph of fcorresponds to an x-intercept on the graph of f′
(because the derivative is then equal to 0). Look for cor-
ners on the graph of f—there will be a jump discontinuity
on the graph of f′. On any interval where fhas a tangent
with positive slope, the graph of f′will be positive (above
the x-axis); similarly, where the tangent has negative slope,
f′is negative. If the graph of fis linear, the graph of f′will
be a horizontal line (because the derivative is then constant).
When comparing your sketch to the one in the answers, keep
in mind that the finer features of the derivative might be
difficult to discern on the graph of f; your sketch doesn’t
necessarily have to be perfect.
Answers
1. y+ 4 = −5(x−2); y+ 4 = 1
5(x−2)
2. (a) m= 1/√a(b) y−2 = 1(x−1) (or y=x+ 1),
y−6 = 1
3(x−9) (or y=1 3x+ 3)
3. −2/a3m/s; −2 m/s; −1
4m/s; −2 27 m/s
4. f(4) = 3, f′(4) = 1
45. f(x) = √x,a= 9
6. f(x) = x6,a= 2
7.
x
y
f′
8.
x
y
f′
10.
x
y
f′
11. f′(t)=4t−5, R,R
12. f′(x) = 1
(1 −4x)2, all x̸=1
4, all x̸=1 4
13. f′(x) = −2x
(x2−4)2, all x̸=±2, all x̸=±2
14. f′(x) = −1
2(1 + x)3/2, (−1,∞), (−1,∞)
15. f′(x) = x
√x2+ 4,R,R
17. a= acceleration, b= velocity, c= position