# Math 171 Review for the Final Exam Solutions

5. Find the values of a and b , if the tangent line to y =
ax^{2} - bx at

(-2, 5) has slope m = 2. (5 points)

**Solution. **Because y(-2) = 5 we have the following equation for a

and b.

4a + 2b = 5

The derivative is 2ax - b. Because we have the second

equation for a and b .

-4a - b = 2.

Adding the equations we get

b = 7

and plugging this value of b into the first equation we get 4a + 14 = 5

whence

6. If a ball is thrown vertically upward with a velocity of 80 ft/s, then its

height after t seconds is h(t) = 80t - 16t^{2}. What is the velocity of the

ball when it is 96 ft above the ground on its way up? On its way

down? (5 points)

**Solution **The velocity of the ball equals to the derivative of its height.

We can find the moments of time when the ball is 96 ft high
by

solving the quadratic equation

80t - 16t^{2} = 96.

Dividing both parts by -16 and moving all terms to the left we get

t^{2} - 5t + 6 = 0

whence t = 2 or t = 3. At the moment t = 2 the velocity of the ball is

16 ft/sec and the ball is moving up whilst at the moment t = 3 the

velocity is -16 ft/sec and the ball is moving down.

7. An open box is to be made from a 6ft by 8ft rectangular piece of

sheet metal by cutting out squares of equal size from the four corners

and bending up the sides. Find the maximum value that the volume

of the box can have.(10 points)

**Solution** Let x be the side of each cut out square. Then the height of

the box is x

whilst its length is 8-2x and its width is 6-2x (See the picture below).

Notice that because these dimensions cannot be negative we
have

0 ≤ x ≤ 3. There fore we have to maximize the volume of the box

V (x) = x(8-2x)(6-2x) = 4x(4-x)(3-x) = 4x(x^{2}-7x+12) = 4(x^{3}-7x^{2}+12x)

on the interval [0, 3]. At the ends of the interval the function V takes

value 0 so it will take the greatest value at a stationary point inside

the interval. To find the stationary points we have to solve the

equation

The quadratic formula provides two solutions

The sign + provides a solution which is greater than 3 and
of no

interest to us. The only stationary point inside the interval [0, 3] is

You can easily check yourself that the greatest value of the volume is

8. Sketch the graph of y = x^{3} - 3x + 2. Find x- and y-
intercepts, plot

the stationary points and the inflection points and determine the

intervals where y is increasing and decreasing, concave up and

concave down.(10 points)

**Solution. **To find the x-intercepts we have to solve the equation

x^{3} - 3x + 2 = 0.

Notice that x = 1 is a solution of this equation and therefore x - 1 is

a factor of x ^{3} - 3x + 2. We can factor x^{3} - 3x + 2 using long division,

grouping , or synthetic division. The table below shows synthetic

division.

Therefore x^{3} - 3x + 2 = (x - 1)(x^{2} + x - 2) = (x - 1)^{2}(x +
2) and the

x- intercepts are (-2, 0) and (1, 0).

The y-intercept is (0, 2).

The first derivative of y is

The function has two stationary points, -1 and 1 which
divide the

x-axis into the intervals (-∞,-1), (-1, 1), and (1,∞). The next

table shows the behavior of the function on these intervals.

Interval | (-∞,-1) | (-1, 1) | (1,∞) |

Sign of | + | - | + |

Behavior of y | Increases | Decreases | Increases |

To find the inflection points of the cubic function y we
compute its

second derivative

We see that y has only one inflection point 0. The
information about

concavity of y is contained in the next table.

Interval | (-1, 0) | (0,1) |

Sign of | - | + |

Concavity | Down | Up |

A computer generated graph of the function y = x^{3} - 3x + 2
is shown

below.

9. Use the fundamental theorem of calculus:(4 points each)

(a) Find the derivative of the function

**Solution.** The fundamental theorem of calculus in its

generalized form tells us that

Applying this formula when a(x) = x^{2}, b(x) = sin x, and

f(t) = sin t^{2} we get

(b) Prove that the function

is a constant on the interval (-∞, 0).

**Solution.** Look at two ways to solve the problem.

First way. We apply the fundamental theorem of calculus to see

that

Because the derivative of F is identically 0 the function
F is a

constant function.

Second way.

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