Multiple Choice
Question 1
The limit lim_{h→0} [cos(π/4 + h) - cos(π/4)] / h represents the derivative of which function at what value?
A
f(x) = sin(x) at x = π/4
B
f(x) = cos(x) at x = 0
C
f(x) = cos(x) at x = π/4
D
f(x) = sin(x) at x = 0
Solution: The limit is the definition f'(a) = lim_{h→0} [f(a+h)-f(a)]/h. Here f(x)=cos(x) and a=π/4.
Question 2
Calculate d/dx [ (x+1)/(x-1) ].
A
2 / (x-1)^2
B
-2 / (x-1)^2
C
1 / (x-1)^2
D
0
Solution: Quotient rule: [(1)(x-1) - (x+1)(1)] / (x-1)^2 = (x-1-x-1) / (x-1)^2 = -2 / (x-1)^2.
Question 3
If f(x) = sin(x), what is the 4th derivative f''''(x)?
A
sin(x)
B
cos(x)
C
-sin(x)
D
-cos(x)
Solution: f' = cos, f'' = -sin, f''' = -cos, f'''' = sin.
Question 4
Estimate the derivative of f(x) at x = 2 using the data: f(1.9) = 10, f(2.1) = 12.
A
5
B
10
C
20
D
1
Solution: Symmetric difference quotient = [f(2.1) - f(1.9)] / (2.1 - 1.9) = (12 - 10) / 0.2 = 2 / 0.2 = 10.
Question 5
Which of the following is equivalent to d/dx [x^2 tan(x)]?
A
2x sec^2(x)
B
x^2 sec^2(x) + 2x tan(x)
C
2x tan(x) + x^2 sec^2(x)
D
Both B and C
Solution: Product rule: d/dx(x^2)tan(x) + x^2(d/dx tan(x)) = 2x tan(x) + x^2 sec^2(x). Both B and C represent this sum.
Question 6
The instantaneous rate of change of y = e^x at x = ln(2) is:
A
1
B
2
C
e
D
ln(2)
Solution: The derivative of e^x is e^x. At x = ln(2), e^{ln(2)} = 2.
Question 7
Using the limit definition, the derivative of f(x) = 2x + 1 at x = 3 is:
A
0
B
1
C
2
D
3
Solution: f'(x) = lim(h→0) [2(x + h) + 1 - (2x + 1)] / h = lim(h→0) (2h / h) = 2.
Question 8
The derivative of f(x) = √x at x = 9 using the limit definition is:
A
1/3
B
1/6
C
1/4
D
1/2
Solution: f'(x) = lim(h→0) [√(x + h) - √x] / h. At x = 9, f'(9) = 1 / (2√9) = 1/6.
Question 9
Given f(2) = 5, f(2.1) = 5.61, and f(1.9) = 4.41, estimate f'(2):
A
1.2
B
3
C
6
D
12
Solution: Symmetric difference quotient: f'(2) ~ (5.61 - 4.41)/0.2 = 1.2/0.2 = 6.
Question 10
The slope of the tangent line to y = cos(x) at x = π is:
A
-1
B
0
C
1
D
π
Solution: The derivative of cos(x) is -sin(x). At x = π, -sin(π) = 0.
Question 11
The derivative of f(x) = x^(-1/2) is:
A
(-1/2)x^(-3/2)
B
(1/2)x^(-3/2)
C
(-1/2)x^(3/2)
D
(1/2)x^(3/2)
Solution: Applying the power rule: d/dx(x^(-1/2)) = (-1/2)x^(-3/2).
Question 12
The derivative of f(x) = cot(x) is:
A
csc²(x)
B
-csc²(x)
C
sec²(x)
D
-sec²(x)
Solution: The derivative of cot(x) is -csc²(x).
Question 13
The derivative of f(x) = (x + 1)(x² - 2x) is:
A
3x² - 2x - 2
B
3x² - 4x - 1
C
3x² - 2x + 1
D
x² - 2x + 1
Solution: Product rule: (1)(x^2 - 2x) + (x + 1)(2x - 2) = x^2 - 2x + 2x^2 - 2 = 3x^2 - 2x - 2.
Question 14
The average rate of change of a function f on the interval [a, b] is represented by which of the following?
A
f'(c) for some c in (a, b)
B
[f(b) - f(a)] / (b - a)
C
lim_{h→0} [f(a+h) - f(a)] / h
D
f(b) - f(a)
Solution: The average rate of change is the slope of the secant line passing through (a, f(a)) and (b, f(b)), which is [f(b) - f(a)] / (b - a).
Question 15
If f(x) = x^2, what is the value of lim_{h→0} [(x+h)^2 - x^2] / h?
A
x
B
2x
C
x^2
D
0
Solution: This is the definition of the derivative of f(x) = x^2. By the power rule, f'(x) = 2x.
Question 16
A function f is differentiable at x = 3 with f(3) = 2 and f'(3) = -4. What is the equation of the line tangent to f at x = 3?
A
y - 2 = -4(x - 3)
B
y + 2 = -4(x + 3)
C
y - 3 = -4(x - 2)
D
y = -4x + 2
Solution: The point-slope form of a line is y - y1 = m(x - x1). Here, (x1, y1) = (3, 2) and m = -4.
Question 17
For which of the following values of x does the graph of f(x) = |x - 5| have a derivative that does not exist?
A
x = 0
B
x = 5
C
x = -5
D
x = 10
Solution: The function f(x) = |x - 5| has a sharp corner (v-shape) at x = 5, where the slope changes abruptly from -1 to 1.
Question 18
Find f'(x) for f(x) = 5x^4 - 2x^2 + 7.
A
20x^3 - 4x
B
5x^3 - 2x
C
20x^3 - 4x + 7
D
20x^5 - 4x^3
Solution: Apply the power rule: d/dx(5x^4) = 20x^3, d/dx(-2x^2) = -4x, and d/dx(7) = 0.
Question 19
If f(x) = 3 sin(x) + 2 cos(x), find f'(x).
A
3 cos(x) + 2 sin(x)
B
-3 cos(x) + 2 sin(x)
C
3 cos(x) - 2 sin(x)
D
-3 cos(x) - 2 sin(x)
Solution: The derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x). Thus, f'(x) = 3 cos(x) + 2(-sin(x)) = 3 cos(x) - 2 sin(x).
Question 20
What is the derivative of f(x) = tan(x) + sec(x)?
A
sec^2(x) + sec(x)tan(x)
B
csc^2(x) + csc(x)cot(x)
C
sec^2(x) - sec(x)tan(x)
D
tan^2(x) + sec(x)
Solution: d/dx(tan x) = sec^2(x) and d/dx(sec x) = sec(x)tan(x).