Deck 28: Graphical Applications of the Derivative

Question 1

Find the equation of the tangent at the given point: $y=x^{3}-2 x^{2}+4$ at $(2,4)$

Answer 1

$y=4 x-4$

Question 2

Tangents to the curve $y^{2}=x$ intersect on the $x$ -axis. The angle between them is $28.072^{\circ}$ . Find the points where the tangents intersect the curve and where they intersect each other.

Answer 2

Tangent - curve $(4,2),(4,-2)$ Each other $(-4,0)$

Question 3

Find the angle of intersection between the curves $y=x^{2}+5 x-8$ and $y=x^{2}-2 x+6$ to three significant digits.

Answer 3

$20.2^{\circ}$

Question 4

Write the equation of the tangent to $x^{2}+y^{2}+2 x y=4$ at the point $(1,1)$ .

Question 5

Find the point(s) on the curve $y=x^{3}-2 x^{2}+4$ at which the tangent is parallel to $y=4 x-23$ .

Question 6

Write the equation of the normal at the given point: $y=3 x^{2}-4$ at $x=2$

Question 7

Find the angle of intersection between the curves: $y=2 x^{3}+x-4$ and $y=2 x^{2}-3$ at $(1,-1)$ to three significant digits

Question 8

Find the angle of intersection between the curves: $y^{2}=4 x$ and $2 x^{2}=12-5 y$ at $(1,2)$ to three significant digits

Question 9

Find the equation of the tangent to the curve $y=\frac{2 x}{1-x^{2}}$ at $x=2$ .

Question 10

Find the equation of the tangent to the curve $y=x^{2}-\sqrt{2 x+1}$ at the point $(4,13)$ .

Question 11

Find the equation of the normal to the curve $y=x^{2}-x^{3}+x$ at the point $(-2,10)$ .

Question 12

Find the equation of the tangent to $y=\sqrt{4 x+1}$ at $(2,3)$ .

Question 13

Find the equation of the normal to $y=4 x(2 x+1)^{3}$ at $(0.5,16)$ .

Question 14

Find the angle of intersection between $y=x^{3}+4 x^{2}$ and $y=x^{2}$ at $(-3,9)$ to three significant digits.

Question 15

Indicate the values where the function is increasing or decreasing: $y=x^{2}-2 x+7$

Question 16

Is the following function increasing or decreasing at the value given? $y=10 x^{2}-5 x+1$ at $x=-1$

Question 17

Is the following function concave up or down at the given $x$ value? $y=2 x^{4}-3 x^{3}+5 x^{2}+7$ at $x=2$

Question 18

Find the maximum, minimum, and inflection points for the equation: $y=2 x^{3}-3 x^{2}-12 x+13$

Question 19

Find the maximum and minimum points for the equation: $y=3 x^{4}+2 x^{3}-3 x^{2}$

Question 20

Find the maximum and the minimum points for the implicit function: $x^{2}+y^{2}-6 x-10 y+30=0$

Question 21

Find the inflection point(s) for the following function: $y=4 x-x^{5}+8$

Question 22

Find the values of $x$ where the function $y=2 x^{2}-x-15$ is increasing and decreasing.

Question 23

Is the function $y=\sqrt[3]{x^{2}-3 x}$ increasing or decreasing at $x=4$ ?

Question 24

Is the function $y=4 x\left(x^{2}+5 x\right)^{2}$ concave up or down at $x=-2$ ?

Question 25

Find the maximum, minimum and the inflection points for the function: $y=2 x^{3}-8 x^{2}-5 x+4$ . Use three significant digits.

Question 26

Find the maximum and the minimum points for the implicit relation: $9 x^{2}+4 y^{2}-18 x-16 y-11=0$

Question 27

Find the maximum, minimum and inflection points for the function: $y=x^{3}-4 x^{2}+5 x-3$ . Use three significant digits.

Question 28

Find the points of inflection for the function: $y=4 x^{4}-6 x^{3}+3 x^{2}$ . Use three significant digits.

Question 29

Find the maximum, minimum, and inflection points for the function: $y=\frac{5 x}{x^{2}+1}$ . Use two decimal places.

Question 30

Find the maximum, minimum and inflection points for the function: $y=6 x^{2}+4\left(x^{2}-1\right)^{2}$ . Use two decimal places.

Question 31

Is the function $y=\frac{x}{x^{2}+1}$ increasing or decreasing at $x=2$ ?

Question 32

Is the function $y=\frac{4 x^{2}}{x-1}$ concave up or down at $x=0.5$ ?

Question 33

Find the maximum, minimum, and inflection points of $y=\frac{2 x}{x^{2}+1}$ . Use two decimal places.

Question 34

Graph the function: $y=\frac{1}{x-2}$

Question 35

Graph the function: $y=\frac{4 x-8}{16 x^{2}-4 x+2}$

Question 36

Make a complete graph of the function: $y=\frac{2}{x^{2}+3}$

Question 37

Make a complete graph of the function: $y=2-\frac{1}{x-4}$

Question 38

Make a complete graph of the function: $y=\frac{x^{2}+x+2}{\sqrt{2 x+1}}$

Question 39

Make a complete graph of function: $y=3 x^{3}-7 x^{2}-5 x+3$

Question 40

Make a complete graph of the function: $y=\frac{x}{\sqrt{4-x^{2}}}$

Question 41

Make a complete graph of the function: $y=2 x^{3}+4 x^{2}-5 x-1$

Question 42

Make a complete graph of the function: $y=8 x^{2}(x-1)^{2}$

Question 43

Make a complete graph of the function: $y=x \sqrt{1-x^{2}}$

Question 44

Make a complete graph of the function: $y=x^{2}+\frac{1}{x^{2}}$

Question 45

Use Newton's method to find the root(s) to two decimal places: $x^{4}-15 x=0$

Question 46

Use Newton's method to find the root(s) to two decimal place: $x^{2}=5$

Question 47

Use Newton's method to find the root(s) to one decimal place: $x^{3}+2 x^{2}=1$

Question 48

Use Newton's method to find the value of the root(s) to two decimal places: $x^{2}=4 x+7$

Question 49

Use Newton's method to find the root(s) of the equation $x^{3}+4 x^{2}=2$ to two decimal places.

Question 50

Use Newton's method to find the root(s) of the equation $x^{2}=2 x+1$ to two decimal places.