Exam 1
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> ?
Let v(x) = f(x) / g(x) . Find v '(−2).
2
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
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)
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
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)
Find the 2021st derivative of f(x) = 2 cos(2x)
−22022sin(2x)
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)
Evaluate lim x→−5+ (2 − x) /(x2 + 4x − 5) .
Find the vector a that has magnitude |a| = 6 and makes an angle of 300◦ with the positive x-axis.
3i − 3 √ 3j
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
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
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
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⟩
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
Evaluate lim x→−∞ √ (9x2 + 12x − 7) /(−2x + 2)
3/2
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
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
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
Find limx→∞ x tan(5/x) ?
5
Find the horizontal and vertical asymptotes for f(x) = (2 − x)(3x + 1) / (x2 − 4)
y = −3, x = −2
Find f'(x) for f(x) = ln (sqrt (x6 + 1))/ sec10 x )? [HINT: Use properties of logarithms.]
(3x5/( x6 + 1)) − 10 tan x
Find the critical numbers of f(x) = x2 /5 (x − 6)2 .
x=0,1,6
Simplify cos(arcsin(x/3)) to an algebraic expression.
sqrt(9 −x2 )/3
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)