Find the point(s) on the graph of \(y = x^2\) at which the tangent line is parallel to the line \(y = 6x - 1\).
A(3, 17)
B(3, 9)
C(1, 2)
D(2, 4)
Mathematics · Question Bank
Differential Calculus
61 of 112 questions match · page 3 of 4. Answer in your head first, then check the solution.
Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7.
A9
B3
C11
D18
Find the point of inflection of the curve \(y = x^3 + 3x^2 - 1\).
A(−1, 1)
B(−2, 3)
C(0, −10)
D(−3, −1)
Find the second derivative of \(y = x\ln x\).
Ax
B1/x
C1
Dx²
The height (in feet) at any time t (in seconds) of a projectile thrown vertically is \(h(t) = -16t^2 + 256t\). What is the projectile’s average velocity for the first 5 seconds of travel?
A48 fps
B96 fps
C176 fps
D192 fps
A ball is thrown from the top of a 1,200-foot building. The position function expressing the height h of the ball above the ground at any time t is \(h(t) = -16t^2 - 10t + 1{,}200\). Find the average velocity for the first 6 seconds of travel.
A−202 ft/sec
B−106 ft/sec
C−96 ft/sec
D−74 ft/sec
Determine the point of inflection of the graph of \(y = x^3 + 6x^2\).
A(−2, 16)
B(0, 0)
C(−1, 5)
D(2, 32)
A rock is dropped down a well that is 256 feet deep. When will it hit the bottom of the well?
A1 sec
B2 sec
C3 sec
D4 sec
If the side of a cube is measured with an error of at most 3 percent, estimate the error in the volume of the cube.
A3 percent
B6 percent
C9 percent
D12 percent
Find the slope of the curve \(x = 3t\), \(y = 9t^2 - 3t\) when t = 1.
A4
B5
C6
D3
Evaluate \(\displaystyle\lim_{x \to 2}\dfrac{x^2 - 4}{x - 2}\).
A4
B6
C8
D16
A runner runs a circular track and a set of data is recorded (time, distance): 68 s, 400 m; 114 s, 600 m; 168 s, 800 m; 209 s, 1,000 m; 256 s, 1,200 m; 322 s, 1,400 m. What is the average velocity from 68 sec to 168 sec?
A3 m/s
B4 m/s
C8 m/s
D6 m/s
A stone, projected vertically upward with initial velocity 112 ft/sec, moves according to \(s = 112t - 16t^2\), where s is the distance from the starting point. Compute the greatest height reached.
A196 ft
B100 ft
C96 ft
D216 ft
Find the value of x for which \(y = 2x^3 - 9x^2 + 12x - 2\) has a maximum value.
A1
B2
C−1
D−2
If f(3) = 7, f′(3) = −2, g(3) = 6 and g′(3) = −10, find \((g/f)'(3)\).
A−82/49
B−49/82
C−49/58
D−58/49
Find the derivative of the function \(y = \dfrac{3}{x^2 + 1}\).
A\(\dfrac{6x}{(x^2 + 1)^2}\)
B\(6x(x^2 + 1)^2\)
C\(\dfrac{-6x}{(x^2 + 1)^2}\)
D\(-6x(x^2 + 1)^2\)
If y = tanh x, find dy/dx.
Asech² x
Bcsch² x
Csinh² x
Dtanh² x
What number exceeds its square by the maximum amount?
A1
B1/2
C1/3
D1/4
Find the derivative of \(x^{-8}\).
A\(-8x^{-9}\)
B\(-8x^{-7}\)
C\(x^{-9}\)
D0
Find the derivative of the function \(y = \dfrac{3}{x^2 + 1}\).
A\(\dfrac{6x}{(x^2 + 1)^2}\)
B\(6x(x^2 + 1)^2\)
C\(\dfrac{-6x}{(x^2 + 1)^2}\)
D\(-6x(x^2 + 1)^2\)