Find the slope of the curve y = x² at x = 2.
A
2
B
4
C
8
D
6
Analysis & Theory
dy/dx = 2x. At x=2 ⇒ slope = 2×2 = 4.
Find the equation of the tangent to y = x³ at x = 1.
A
y - 1 = 3(x - 1)
B
y - 1 = (x - 1)
C
y - 1 = 2(x - 1)
D
y = x³
Analysis & Theory
dy/dx = 3x² ⇒ slope at x=1 is 3. Point is (1,1). Tangent: y-1=3(x-1).
Find the equation of the normal to y = x² at x = 1.
A
y - 1 = -1/2(x - 1)
B
y - 1 = -(x - 1)/2
C
y - 1 = -1/2(x - 1)
D
y - 1 = 2(x - 1)
Analysis & Theory
Slope of tangent = 2x ⇒ at x=1 slope = 2 ⇒ slope of normal = -1/2 ⇒ equation: y-1 = -1/2(x-1).
For y = x³ - 3x, find the x-coordinate where slope = 0.
A
0
B
±1
C
√3
D
No solution
Analysis & Theory
dy/dx = 3x² - 3 = 0 ⇒ x²=1 ⇒ x=±1.
Find the intervals where y = x² + 2x + 3 is increasing.
A
(-∞, -1)
B
(-1, ∞)
C
(0, ∞)
D
(-∞, ∞)
Analysis & Theory
dy/dx = 2x+2 > 0 ⇒ x > -1. Hence function is increasing for (-1, ∞).
Find the intervals where y = x² + 2x + 3 is decreasing.
A
(-∞, -1)
B
(-1, ∞)
C
(0, ∞)
D
(-∞, ∞)
Analysis & Theory
dy/dx = 2x+2 < 0 ⇒ x < -1. Hence function is decreasing for (-∞, -1).
For y = x³, find whether the function is increasing or decreasing at x = 0.
A
Increasing
B
Decreasing
C
Neither
D
Constant
Analysis & Theory
dy/dx = 3x² ≥ 0 for all x ⇒ function is increasing everywhere.
Find the maximum value of y = 6x - x².
A
9
B
6
C
3
D
0
Analysis & Theory
dy/dx = 6 - 2x = 0 ⇒ x=3. y(3)=6(3)-9=9 ⇒ maximum value = 9.
Find the minimum value of y = x² + 4x + 5.
A
1
B
5
C
-4
D
0
Analysis & Theory
dy/dx = 2x+4=0 ⇒ x=-2 ⇒ y(-2)=(-2)²+4(-2)+5=1 ⇒ minimum value = 1.
If slope of a curve at point P is m, what is the slope of the normal at P?
A
m
B
-1/m
C
1/m
D
-m
Analysis & Theory
Slope of normal = negative reciprocal of slope of tangent ⇒ slope = -1/m.