Math final

Exam 1

100

Find a vector of length 3 in the same direction as the vector from the point (−2, 3) to (1, −1)

?< 9/5> , <− 12/ 5> ?

200

A crate is hauled 10 meters up a ramp under a constant force of 7 Newtons applied at an angle of 30◦ to the ramp. Find the work done.

35sqrt(3)

300

Given f(x) =x³ − 4x + 1 and f ′ (x) = 3x² − 4, find the equation of the tangent line to f(x) at x = −2.

y = 8x+17

400

Evaluate lim x→−∞ √ (9x² + 12x − 7) /(−2x + 2)

3/2

500

Find the horizontal and vertical asymptotes for f(x) = (2 − x)(3x + 1) / (x² − 4)

y = −3, x = −2

Exam 2

100

Let v(x) = f(x) / g(x) . Find v '(−2).

2

200

Find the 2021st derivative of f(x) = 2 cos(2x)

−2²⁰²²sin(2x)

300

A ball is tossed in the air, and the height of the ball at time t seconds is given by h(t) = 25 + 10t − t² , where h(t) is measured in feet from the ground. Find the maximum height H of the ball and the time T when it hits the maximum height

H = 50, T = 5

400

There are two lines tangent to the parabola f(x) = x² + 1 that pass through the point (1, −2). Find the x-coordinates where these tangent lines touch the parabola.

x = −1, 3

500

Find f'(x) for f(x) = ln (sqrt (x⁶ + 1))/ sec¹⁰ x )? [HINT: Use properties of logarithms.]

(3x⁵/( x⁶ + 1)) − 10 tan x

Exam 3

100

Find the most general antiderivative of the function f(x) = (8x − 2x³sec(x) tan(x) + 6x²)/ x³.

F(x) = (-8x^-1) − 2 sec(x) + 6 ln |x| + C

200

Given that f(x) is defined everywhere except x = −3 and f'(x) = x²(1 − x) /(x + 3)³, on what intervals is f(x) increasing?

(-3,1)

300

An object is traveling at a speed of 60 m/s when the brakes are fully applied, producing a constant deceleration of 12 meters per second squared. What is the distance covered before the object comes to a stop?

150m

400

Estimate the area under the graph of f(x) = x²+2 from x = −3 to x = 6 using three rectangles of equal width and left endpoints.

72

500

Find the critical numbers of f(x) = x² /5 (x − 6)² .

x=0,1,6

ooooo mystery

100

Let g(x) = f(x² + 1). Given the table of values below for f and f' , find the equation of the tangent line to g(x) at x = 1.

x.  f(X).  f'(x)

1.   3.    4

2.    5.    6

3.     1.   2

y − 5 = 12(x − 1)

200

Evaluate lim x→−5⁺ (2 − x) /(x² + 4x − 5) .

- infinity

300

Find a vector equation of the line that passes through (−6, 4) and is perpendicular to the line with parametric equations x = 7 + 2t, y = 1 − 3t.

r(t) = ⟨−6 + 3t, 4 + 2t⟩

400

Find the slope of the tangent line at the point (2, 0) for the following parametric curve x(t) =t⁴ + 1, y(t) = cos ?(πt /2) ?

- pi/4

500

Simplify cos(arcsin(x/3)) to an algebraic expression.

sqrt(9 −x² )/3

Oooooo mystery

100

The radius of a circle was measured to be 5 ft with a maximum possible error of 0.2 ft. Use differentials to estimate the maximum possible error in the calculated area of the circle.

2pi

200

Find the vector a that has magnitude |a| = 6 and makes an angle of 300◦ with the positive x-axis.

3i − 3 √ 3j

300

Find the absolute maximum and minimum values of f(x) = 6sqrt(x) − x + 1 on [0, 25].

Absolute maximum is 10; absolute minimum is 1

400

Find limx→∞  x tan(5/x) ?

5

500

Find all point(s) on the curve parametrized by 

x = x² − 2t − 3, 

y =t³ − 3t ² where the tangent line is vertical.

(-4,-2)