> WQStools[makehead](`123/homework/chap_6_homework`); 
BEGINWQSHEADER
 
E1:
 
F1:/u/disk13/ma/carl/public_html/dolts/123/homework/chap_6_homework
 
U1:http://www.ms.uky.edu/~carl/dolts/123/homework/chap_6_homework
 
ENDWQSHEADER
T1
T1_
Differentiation Formulas

In the following formulas, f(x) and g(x) are differentiable functions and c is a constant. All but the last should already be familiar.

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math] (x) = [Maple Math]

[Maple Math]

[Maple Math] if [Maple Math] is the inverse of [Maple Math]


Formulas for "calculator button" functions.

The table below contains the derivatives for most of the commonly used functions represented by buttons on your calculator. We will study such things as the fact that the derivative of tan(x) is [Maple Math] later in the course. However in the mean time we will use the facts in this table in derivative formulas whenever necessary to calculate derivatives such as the ones below.


Note that Maple writes [Maple Math] when your text writes [Maple Math] or [Maple Math] f


[Maple Math]
SKIP
code for table
> MATRIX([[f(x), diff(f(x),x), f(x), diff(f(x),x)], [x^a, a*x^(a-1), a^x, a^x*ln(a)], [ln(x), 1/x, Log[a](x), diff(Log[a](x),x)], [sin(x), cos(x), cos(x), -sin(x)], [tan(x), (sec^2)(x), cot(x), -(csc^2)(x)], [sec(x), sec(x)*tan(x), csc(x), -csc(x)*cot(x)], [arcsin(x), 1/sqrt(1-x^2), arccos(x), -1/sqrt(1-x^2)], [arctan(x), 1/(1+x^2), arccot(x), -1/(1+x^2)], [arcsec(x), 1/(x^2*sqrt(1-1/(x^2))), arccsc(x), -1/(x^2*sqrt(1-1/(x^2)))], [sinh(x), cosh(x), cosh(x), sinh(x)]]);
[Maple Math]
> makequest(1,4);
Q1_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q1: {include:Q1_.html}
T1:{include:T1_1.html}
R1: A2
A1: {include:Q1_A1_.html}
A2: {include:Q1_A2_.html}
A3: {include:Q1_A3_.html}
A4: {include:Q1_A4_.html}
ENDWQSLINE
calculations
> makequest(2,4);
Q2_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_ ANSWER
[Maple Math]
BEGINWQSLINE
Q2: {include:Q2_.html}
R2: A4
A1: {include:Q2_A1_.html}
A2: {include:Q2_A2_.html}
A3: {include:Q2_A3_.html}
A4: {include:Q2_A4_.html}
ENDWQSLINE
>


> makequest(3,4);
Q3_
[Maple Math]
A1_ ANSWER
[Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q3: {include:Q3_.html}
R3: A1
A1: {include:Q3_A1_.html}
A2: {include:Q3_A2_.html}
A3: {include:Q3_A3_.html}
A4: {include:Q3_A4_.html}
ENDWQSLINE
>

> makequest(4,4);
>
calculations
Q4_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_ ANSWER
[Maple Math]
A4_
( [Maple Math]
BEGINWQSLINE
Q4: {include:Q4_.html}
R4: A3
A1: {include:Q4_A1_.html}
A2: {include:Q4_A2_.html}
A3: {include:Q4_A3_.html}
A4: {include:Q4_A4_.html}
ENDWQSLINE
>
5. [Maple Math]

> makequest(5,4);
>
calculations
Q5_
f(x) = [Maple Math]
A1_
[Maple Math]
A2_ ANSWER
 [Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q5: {include:Q5_.html}
R5: A2
A1: {include:Q5_A1_.html}
A2: {include:Q5_A2_.html}
A3: {include:Q5_A3_.html}
A4: {include:Q5_A4_.html} 
ENDWQSLINE
>


> makequest(6,4);
>
calculations
Q6_
f(x) = [Maple Math]
A1_ ANSWER
 [Maple Math]

A2_
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q6: {include:Q6_.html}
R6: A1
A1: {include:Q6_A1_.html}
A2: {include:Q6_A2_.html}
A3: {include:Q6_A3_.html}
A4: {include:Q6_A4_.html}
ENDWQSLINE
>
>

> makequest(7,4);
>
calculations
Q7_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q7: {include:Q7_.html}
R7: A2
A1: {include:Q7_A1_.html}
A2: {include:Q7_A2_.html}
A3: {include:Q7_A3_.html}
A4: {include:Q7_A4_.html}
ENDWQSLINE
>

8. [Maple Math]

> makequest(8,4);
>
calculations
Q8_
[Maple Math]
A1_
[Maple Math]
A2_
 [Maple Math]

A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q8: {include:Q8_.html}
R8: A2
A1: {include:Q8_A1_.html}
A2: {include:Q8_A2_.html}
A3: {include:Q8_A3_.html}
A4: {include:Q8_A4_.html}
ENDWQSLINE
>
calculations
>

9. [Maple Math]

> makequest(9,4);
Q9_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_ ANSWER
 [Maple Math]

BEGINWQSLINE
Q9: {include:Q9_.html}
R9: A4
A1: {include:Q9_A1_.html}
A2: {include:Q9_A2_.html}
A3: {include:Q9_A3_.html}
A4: {include:Q9_A4_.html}
ENDWQSLINE
>
>
11. [Maple Math]
> makequest(11,4);
>
calculations
Q11_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_ ANSWER
 [Maple Math]

BEGINWQSLINE
Q11: {include:Q11_.html}
R11: A4
A1: {include:Q11_A1_.html}
A2: {include:Q11_A2_.html}
A3: {include:Q11_A3_.html}
A4: {include:Q11_A4_.html}
ENDWQSLINE
>
>
12. [Maple Math]

> makequest(12,4);
>
calculations
> f:= sin(x^2); ans:=diff(f,x);
[Maple Math]
[Maple Math]
>
> f:=sin(-x^2);diff(f,x);
[Maple Math]
[Maple Math]
Q12_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
 [Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q12: {include:Q12_.html}
R12: A2
A1: {include:Q12_A1_.html}
A2: {include:Q12_A2_.html}
A3: {include:Q12_A3_.html}
A4: {include:Q12_A4_.html}
ENDWQSLINE
>
calculations



22. [Maple Math]

> makequest(13,4);
>
Q13_
[Maple Math]
A1_ ANSWER
 [Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q13: {include:Q13_.html}
R13: A1
A1: {include:Q13_A1_.html}
A2: {include:Q13_A2_.html}
A3: {include:Q13_A3_.html}
A4: {include:Q13_A4_.html}
ENDWQSLINE
>


23. [Maple Math]

>
> makequest(14,4);
>
calculations
Q14_
[Maple Math]
A1_
[Maple Math]

A2_
[Maple Math]
A3_ ANSWER
 [Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q14: {include:Q14_.html}
R14: A3
A1: {include:Q14_A1_.html}
A2: {include:Q14_A2_.html}
A3: {include:Q14_A3_.html}
A4: {include:Q14_A4_.html}
ENDWQSLINE
>


24. [Maple Math] [Maple Math]
> makequest(15,4);
>
calculations
Q15_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]

BEGINWQSLINE
Q15: {include:Q15_.html}
R15: A2
A1: {include:Q15_A1_.html}
A2: {include:Q15_A2_.html}
A3: {include:Q15_A3_.html}
A4: {include:Q15_A4_.html}
ENDWQSLINE
> makequest(16,4);
>
calculations
> f:= sin(x)/sqrt(1+x); ans:=diff(f,x);
[Maple Math]
[Maple Math]
>
> f:=sin(x)/sqrt(1+x);diff(sin(x),x)*diff(sqrt(1+x),x);
[Maple Math]
[Maple Math]
> -3*cos(x)^2;
> -3*cos(x)^2;
Q16_
[Maple Math]
A1_ ANSWER
 [Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q16: {include:Q16_.html}
R16: A1
A1: {include:Q16_A1_.html}
A2: {include:Q16_A2_.html}
A3: {include:Q16_A3_.html}
A4: {include:Q16_A4_.html}
ENDWQSLINE
> makequest(17,4);
>
calculations
>
Q17_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q17: {include:Q17_.html}
R17: A2
A1: {include:Q17_A1_.html}
A2: {include:Q17_A2_.html}
A3: {include:Q17_A3_.html}
A4: {include:Q17_A4_.html}
ENDWQSLINE
> makequest(18,4);
>
calculations
Q18_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_ ANSWER
 [Maple Math]

A4_
[Maple Math]
BEGINWQSLINE
Q18: {include:Q18_.html}
R18: A3
A1: {include:Q18_A1_.html}
A2: {include:Q18_A2_.html}
A3: {include:Q18_A3_.html}
A4: {include:Q18_A4_.html}
ENDWQSLINE
> makequest(19,4);
>
calculations
>
Q19_
[Maple Math]
A1_
[Maple Math]
A2_ ANSWER
 [Maple Math]

A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q19: {include:Q19_.html}
R19: A2
A1: {include:Q19_A1_.html}
A2: {include:Q19_A2_.html}
A3: {include:Q19_A3_.html}
A4: {include:Q19_A4_.html}
ENDWQSLINE
> makequest(20,4);
>
calculations
Q20_
[Maple Math]
A1_
[Maple Math]
A2_
[Maple Math]
A3_
[Maple Math]
A4_ ANSWER
 [Maple Math]

BEGINWQSLINE
Q20: {include:Q20_.html}
R20: A4
A1: {include:Q20_A1_.html}
A2: {include:Q20_A2_.html}
A3: {include:Q20_A3_.html}
A4: {include:Q20_A4_.html}
ENDWQSLINE
> makequest(21,4);
Q21_
Suppose f(x) and g(x) are functions such that:
a. [Maple Math]
b. [Maple Math]
c. [Maple Math]
d. [Maple Math]
Find [Maple Math] and [Maple Math] [Maple Math] (3)
A1_ ANSWER
10, 60
A2_
-6,60
A3_
10, 36
A4_
-6, 36
BEGINWQSLINE
Q21: {include:Q21_.html}
R21: A1
A1: {include:Q21_A1_.html}
A2: {include:Q21_A2_.html}
A3: {include:Q21_A3_.html}
A4: {include:Q21_A4_.html}
ENDWQSLINE
> makequest(22,4);
Q22_

Suppose f(x) and g(x) are functions such that:
a [Maple Math]
b. [Maple Math]
c. [Maple Math]
d. [Maple Math]
Find [Maple Math] and [Maple Math] [Maple Math] (3)
A1_
10, 12
A2_
32, -12
A3_ ANSWER
-8, 12

A4_
12, -10
BEGINWQSLINE
Q22: {include:Q22_.html}
R22: A3
A1: {include:Q22_A1_.html}
A2: {include:Q22_A2_.html}
A3: {include:Q22_A3_.html}
A4: {include:Q22_A4_.html}
ENDWQSLINE
> makequest(23,4);
Q23_

Suppose f(x) and g(x) are functions such that:
a. [Maple Math]
b. [Maple Math]
c. [Maple Math]
d. [Maple Math]
Find ( [Maple Math] [Maple Math] (2)
A1_
[Maple Math]
A2_
 [Maple Math]

A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q23: {include:Q23_.html}
R23: A2
A1: {include:Q23_A1_.html}
A2: {include:Q23_A2_.html}
A3: {include:Q23_A3_.html}
A4: {include:Q23_A4_.html}
ENDWQSLINE
> makequest(24,4);
Q24_

Suppose f(x) and g(x) are functions such that:
a. [Maple Math]
b. [Maple Math]
c. [Maple Math]
d. [Maple Math]

What is the equation of the tangent line to the graph of [Maple Math] at x =-3?
A1_ ANSWER
 [Maple Math]

A2_
y [Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q24: {include:Q24_.html}
R24: A1
A1: {include:Q24_A1_.html}
A2: {include:Q24_A2_.html}
A3: {include:Q24_A3_.html}
A4: {include:Q24_A4_.html}
ENDWQSLINE
> makequest(25,4);
> h:=x->2/3 + 13/9*(x-1);
[Maple Math]
> h:=x->2/3 - 29/9*(x-1);
[Maple Math]
>
> h(1.2);
[Maple Math]
> .3777
>
Q25_

Suppose f(x) and g(x) are functions such that:
a. [Maple Math]
b. [Maple Math]
c. [Maple Math]
d. [Maple Math]
Use the tangent line approximation to estimate [Maple Math]
A1_
1.3111
A2_
 .9555

A3_
.3777
A4_
.7500
BEGINWQSLINE
Q25: {include:Q25_.html}
R25: A2
A1: {include:Q25_A1_.html}
A2: {include:Q25_A2_.html}
A3: {include:Q25_A3_.html}
A4: {include:Q25_A4_.html}
ENDWQSLINE
> makequest(26,4);
> F:=x^2-x*y(x)+y(x)^3 = 8;
[Maple Math]
> df:=diff(F,x);
[Maple Math]
> solve(%,diff(y(x),x));
[Maple Math]
> df := 2*x-y(x)-x*diff(y(x),x)+3*y(x)^2 = 0;
>
[Maple Math]
>
>
Q26_
Suppose y is a function of x ( y = y(x)) such that [Maple Math] . Use implicit differentiation on the following to calculate [Maple Math]
A1_
[Maple Math]
A2_ ANSWER
[Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q26: {include:Q26_.html}
R26: A2
A1: {include:Q26_A1_.html}
A2: {include:Q26_A2_.html}
A3: {include:Q26_A3_.html}
A4: {include:Q26_A4_.html}
ENDWQSLINE
> makequest(27,4);
>
> F:=y(x)^2+y(x)+x^2 -7;
[Maple Math]
> df:=diff(F,x);
[Maple Math]
> solve(df,diff(y(x),x));
[Maple Math]
>
> df := 2*y(x)+diff(y(x),x)+2*x;
>
>
[Maple Math]
>
>
>
Q27_

Suppose y is a function of x ( y = y(x)) such that [Maple Math] . Use implicit differentiation on the following to calculate [Maple Math]
A1_
[Maple Math]
A2_ ANSWER
 [Maple Math]
A3_
[Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q27: {include:Q27_.html}
R27: A2
A1: {include:Q27_A1_.html}
A2: {include:Q27_A2_.html}
A3: {include:Q27_A3_.html}
A4: {include:Q27_A4_.html}
ENDWQSLINE
> makequest(28,4);
>
>
>
Q28_
Suppose (2,1) is a point on the graph of the function y=f(x) where f is defined implicitly by [Maple Math] . What is the slope of the tangent line to the graph of y =f(x) at the point (2,1)? What is the equation of the tangent line? What is the tangent line approximation to f(2.3)?
A1_
[Maple Math]
A2_
 [Maple Math]

A3_
- [Maple Math]
A4_
[Maple Math]
BEGINWQSLINE
Q28: {include:Q28_.html}
R28: A2
A1: {include:Q28_A1_.html}
A2: {include:Q28_A2_.html}
A3: {include:Q28_A3_.html}
A4: {include:Q28_A4_.html}
ENDWQSLINE
>