## Derivatives

1. What is a derivative?

2. When looking at a graph of a function, what does the derivative at a specific point represent?

3. What are derivatives useful in determining?

4. Using the derivative rules determine

4a.
**
5
x
2
{\displaystyle 5x^{2}}
**

4b.
**
5
x
2
+
6
x
5
+
17
{\displaystyle 5x^{2}+6x^{5}+17}
**

4c.
**
(
3
x
2
+
5
)
(
9
x
7
−
9
)
{\displaystyle (3x^{2}+5)(9x^{7}-9)}
**

4d.
**
(
l
n
x
+
7
x
4
+
x
5
)
(
7
x
7
+
2
x
5
−
9
)
{\displaystyle (lnx+7x^{4}+x^{5})(7x^{7}+2x^{5}-9)}
**

4e.
5
x
2
−
8
x
−
13
x
2
−
5
{\displaystyle {\frac {5x^{2}-8x-13}{x^{2}-5}}}

4f.
7
x
5
−
8
x
5
−
13
x
7
x
3
−
5
{\displaystyle {\frac {7x^{5}-8x^{5}-13x}{7x^{3}-5}}}

4g.
**
(
7
x
7
+
2
x
5
−
9
)
3
{\displaystyle (7x^{7}+2x^{5}-9)^{3}}
**

4h.
7
x
7
+
2
x
5
−
9
{\displaystyle {\sqrt {7x^{7}+2x^{5}-9}}}

4i.
**
s
i
n
(
5
x
)
{\displaystyle sin(5x)}
**

4j.
9
x
3
−
4
x
2
−
4
x
2
6
x
3
−
10
{\displaystyle {\sqrt {\frac {9x^{3}-4x^{2}-4x^{2}}{6x^{3}-10}}}}

## Limits

1. What is a limit?

2. Using limits determine

2a.
lim
x
→
2
2
x
+
1
{\displaystyle \lim _{x\to 2}2x+1}

2b.
lim
x
→
3
5
x
2
−
8
x
−
13
x
2
−
5
{\displaystyle \lim _{x\to 3}{\frac {5x^{2}-8x-13}{x^{2}-5}}}

2c.
lim
x
→
1
x
3
−
1
(
x
−
1
)
2
{\displaystyle \lim _{x\to 1}{\frac {x^{3}-1}{(x-1)^{2}}}}

2d.
lim
x
→
4
3
−
(
x
+
5
)
x
−
4
{\displaystyle \lim _{x\to 4}{\frac {3-{\sqrt {(x+5)}}}{x-4}}}

2e.
lim
x
→
3
x
4
−
81
2
x
2
−
5
x
−
3
{\displaystyle \lim _{x\to 3}{\frac {x^{4}-81}{2x^{2}-5x-3}}}

3. What are the two variations of the definition of the derivative?

4. Using the Definition of the derivative determine

4a.
**
5
x
2
{\displaystyle 5x^{2}}
**

4b.
**
5
x
2
−
3
x
−
7
{\displaystyle 5x^{2}-3x-7}
**

4c. 4 − x + 3 {\displaystyle 4-{\sqrt {x+3}}}

## Continuity

1. What makes a function continuous.

2. If a function is differentiable at a point is it continuous?

3. If a function is continuous at a point is it differentiable?

## Implicit Differentiation

1. What makes implicit differentiation problems from standard differentiation problems?

2. Using implicit differentiation methods determine **dy/dx** for the following

2a.
**
x
3
+
y
3
=
7
{\displaystyle x^{3}+y^{3}=7}
**

2b.
**
(
x
−
y
)
2
=
x
+
y
+
2
{\displaystyle (x-y)^{2}=x+y+2}
**

2c.
**
x
3
y
5
+
x
3
y
2
=
y
{\displaystyle x^{3}y^{5}+x^{3}y^{2}=y}
**

2d.
**
x
3
+
y
3
=
7
{\displaystyle x^{3}+y^{3}=7}
**

2d.
**
e
4
x
−
e
2
x
=
y
{\displaystyle e^{4x}-e^{2x}=y}
**

2e.
x
2
=
x
2
+
y
2
{\displaystyle x^{2}={\sqrt {x^{2}+y^{2}}}}

2f.
4
x
2
+
y
3
y
2
−
5
x
{\displaystyle {\frac {4x^{2}+y}{3y^{2}-5x}}}

## Related Rates

**This are a few Problems from past finals courtesy of the Math Exam Resources (MER)**

1a. Shadows Question

1b. Shadows Question

**This is a PDF courtesy of Utah Education. I has a few different styles of problems.**

## Optimization

**Independent Problems**

1. We need to enclose a field with a fence. We have 500 meters of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the dimensions of the field that will enclose the largest area.

2. We want to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost \$ 10/meters^2 and the material used to build the sides cost \$ 6/meters^2. If the box must have a volume of 50meters^3 determine the dimensions that will minimize the cost to build the box.

3. We want to construct an equal sided box with a square base and we only have 10 m^2 of material to use in construction of the box. Assuming that all the material is used in the construction process determine the maximum volume that the box can have.

4. A manufacturer needs to make a cylindrical can that will hold 1.5 m^3 of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction.

5. A printer need to make a poster that will have a total area of 200 cm2 and will have 1 cm margins on the sides, a 2 cm margin on the top and a 1.5 cm margin on the bottom. What dimensions will give the largest printed area?

**Problems from the Math Text Book**

Calculus Early Transcendentals, Volume 1, Third Edition

Page. 261-267

**Problems from the Math Exam Resources Wiki**