November 20, 2016
MATH 255A
Prof. V. Panferov
Midterm 3: Study Guide
Textbook coverage:
3.3: Chain Rule; Implicit Differentiation; Derivatives of Logarithms. Examples 1, 2, 3, 5.
3.4: Derivative Rules for Sine and Cosine; Derivative Rules for Other Trigonometric Functions;
Examples 2, 3, 4.
3.5: Linear Approximation; Error Estimates, Sensitivity and Elasticity; Examples 3, 7, 8, 9.
3.6: Higher derivatives; Concave Up/Concave Down; Quadratic Approximation; Examples 3, 6,
7, 8.
4.1: Graphing with Calculus; Examples 2 (problem 15), 3 (problem 6), 4 (problem 31).
4.2: Local Maxima and Minima; Fermat’s theorem; Critical Points; First and Second Derivative
Tests; Global Extrema; Closed and Open Interval Methods; Examples 2, 3, 4, 6, 9 (problem
24).
4.3: Steps to Solve an Optimization Problem; Examples 1, 3, 5.
List of Review Questions
1. Find
dy
dx
for the following expressions:
(a) y =
√
√
(c) y = x2 e− x
√
(d) y = x x cos(2x)
1
5−x+x4
(b) y = log2 (2x + 5)
(e) xy + y 3 = 25
(f) y = sin2 ( πx
).
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2. Use the linear approximation of the function y = x1/3 to estimate the value 651/3
and determine whether this approximation overestimates or underestimates the true
answer. Justify your answers by using derivatives.
2
3. Find the linear and quadratic approximations to the function y = e−x at x =
Graph the function and its approximations.
√
2
.
2
4. (a) Find the sensitivity of y = f (x) at the point x = a and use it to estimate ∆y for
the given measurement error ∆x:
y=
√
2x2 + 1,
1
a = −2,
∆x = 0.01.
(b) Find the elasticity of y = f (x) at the point x = a and use it to estimate the
percent error in y for the given percent error in x:
y=
√
2x2 + 1,
a = −2,
with 8% error in x.
5. A certain cell is modeled as a sphere. If the formulas S = 4πr2 and V = 43 πr3 are used
to compute the surface area and volume of the sphere, respectively, estimate the effect
on S and V produced by a 1% increase in the radius r. Use the linear approximation.
6. A bacterial colony is estimated to have a population of P thousand individuals, where
P (t) =
24t + 10
,
t2 + 1
and t is the number of hours after a toxin is introduced.
(a) At what rate is the population changing when t = 0 and t = 1?
(b) Is the rate increasing or decreasing at t = 0 and t = 1?
(c) At what time does the population begin to decrease?
7. Suppose the concentration in the blood at time t of a drug injected into the body is
modeled by
C(t) = te−2t .
)n (it follows from the definition of the number e)
Using the inequality e2t > (1 + 2t
n
show that the function has a horizontal asymptote. Find the time when C 0 (t) = 0.
Graph this curve and verify that the largest concentration occurs at the solution to
C 0 (t) = 0.
8. Let f be a function defined by
y = x3 + 35x2 − 125x − 9375.
Determine where the function is increasing, where it is decreasing, and where the
graph is concave up or concave down.
9. Consider the following approximate function for the blood pressure of a patient:
P (t) = 100 + 20 cos
where t is measured in minutes.
2
π(t − 15) 35
mmHg
(a) Find P 0 (20) and interpret.
(b) During what periods of the first hour is the blood pressure increasing?
(c) At what time does the blood pressure achieve its minumum?
10. Sketch a graph of a function y = f (x) with the following properties:
f 0 (x) > 0 when x < 1;
f 0 (x) < 0 when x > 1;
f 00 (x) > 0 when x < 1;
f 00 (x) > 0 when x > 1.
What happens with the derivative of f when x = 1?
11. Sketch the graph of a function y = f (x) with the following properties:
y = ±1 are horizontal asymptotes;
f (x) is decreasing on (−∞, ∞);
f (x) is concave down for x < −1 and for 0 < x < 1;
f (x) is concave up for −1 < x < 0 and for x > 1.
12. Find the global maximum and the global minimum of f (x) =
√
xe−x on [0, 6].
13. A model using the function
C(t) =
k
(e−at − e−bt )
b−a
has been suggested for describing the concentration in the blood of a drug injected
into the body intramuscularly. Here a, b and k are positive constants, with b > a. At
what time does the largest concentration occur? What happens to the concentration
as t → ∞?
14. During the winter, a species of bird migrates from the coast of a mainland to an island
500 miles southeast. If the energy the bird requires to fly one mile over the water is
50% more than the amount of energy it requires to fly over the land, determine what
path the species should fly to minimize the amount of energy used. (Assume that the
coast line runs from north to south.)
For other types of review questions look up problems similar to ones in online homework.
Examples: 4.3: 2, 9, 13, 15; 4.2: 9, 19, 23; 4.1: 3, 5, 33, 35; 3.6: 23, 24, 28, 29, 43, 50;
3.5: 15, 21, 27, 30; 3.4: 17, 19, 27, 35; 3.3: 11, 19, 21, 43.
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Answers:
1. (a) 12 (1 − 4x3 )(5 − x + x4 )−3/2 , hint: write the function as y = (5 − x + x4 )−1/2 ; (b)
√ −√x
2
1
,
hint:
log
x
=
(ln
x)/(ln
2)
(change
of
base
formula);
(c)
x
(4
−
x)e
;
2
(ln 2)(2x+5)
2
√
y
1√
3/2
0
(d) 2 x (3 cos(2x) − 4x sin(2x)), hint: write x x as x ; (e) y = − x+3y2 hint: use
; hint: trig identity sin(2x) = 2 sin x cos x helps
implicit differentiation; (f) π4 sin πx
2
simplify the answer.
2. 4 + 1/48 ≈ 4.02083; this is an overestimate. Use the linear approximation x1/3 ≈
1
4 + 48
(x − 64); since the second derivative (x1/3 )00 = − 92 x−5/3 is always negative, the
function is concave down and the graph lies below the tangent line.
√
√
3. y = 2e−1/2 − 2e−1/2 x; the second derivative is zero at x = 22 so the quadratic
approximation concides with the linear one.
4. (a) f 0 (−2) = − 43 , ∆y ≈ −0.0133; (b) E = 98 , 7.1%.
5. Area increases by ≈ 2%; volume increases by ≈ 3%.
6. (a) P 0 (0) = 24; P 0 (1) = −5; (b) decreasing at both t = 0 and t = 1; (c) t = 23 .
7. Horizontal asymptote y = 0; maximum occurs at t = 21 .
8. Increases on (−∞, −25) and on ( 53 , ∞); decreases on (−25, 35 ); concave down on
(−∞, − 35
), concave up on (− 35
, ∞).
3
3
sin π7 ≈ −0.77891. The blood presure is decreasing after 20 minutes,
9. (a) P 0 (20) = − 4π
7
at a rate ≈ 0.78 mmHg/min; (b) increasing on (0, 15) and (50, 60); decreasing on
(15, 50); (c) the global minumum is reached at t = 50.
p
10. One possible function is y = − |x − 1|. The function is necessarily not differentiable
at x = 1.
12. Global maximum
13. t =
ln b−ln a
.
b−a
√1 ,
2e
achieved at x = 12 ; global minimum 0, achieved at x = 0.
The concentration decays to zero.
14. 37.3 miles along the coast, then straight to the island.
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