1
What is the derivative of the product of two functions, and ?
Correct Answer:
Explanation: According to the product rule of differentiation, .
2
What is the derivative of with respect to ?
Correct Answer:
Explanation: The standard derivative of is .
3
If , find .
Correct Answer:
Explanation: Using the chain rule, .
4
If and , what is ?
Correct Answer:
Explanation: , . .
5
Find the derivative of the implicit function .
Correct Answer:
Explanation: Differentiating w.r.t : .
6
Logarithmic differentiation is most useful for functions of the form:
Correct Answer:
Explanation: Logarithmic differentiation is primarily used when the base and exponent are both functions of .
7
If , what is ?
Correct Answer:
Explanation: Taking log: . Differentiating: . Thus .
8
Which of the following is a property of the indefinite integral?
A.
B.
C.
D. None of these
Correct Answer:
Explanation: Constants can be taken out of the integral sign. This is the constant multiple rule.
9
What is the formula for integration by parts?
Correct Answer:
Explanation: The standard formula for integration by parts is derived from the product rule of differentiation: .
10
In the integration by parts method using the ILATE rule, what does 'L' stand for?
A. Linear
B. Logarithmic
C. Limit
D. Long
Correct Answer: Logarithmic
Explanation: ILATE stands for Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. It helps in choosing .
11
What is the correct partial fraction form for ?
Correct Answer:
Explanation: Since the denominator consists of distinct linear factors, the decomposition is .
12
Evaluate if is an odd function.
Correct Answer: $0$
Explanation: For an odd function, the area under the curve from to $0$ cancels out the area from $0$ to . Thus, the integral is 0.
13
What is the derivative of ?
Correct Answer:
Explanation: Using chain rule: .
14
Find .
Correct Answer:
Explanation: The standard derivative of an exponential function with base is .
15
If and , find .
Correct Answer:
Explanation: . .
16
Differentiate .
Correct Answer:
Explanation: The standard derivative of is .
17
If , find .
Correct Answer:
Explanation: Differentiating product : .
18
Evaluate for .
Correct Answer:
Explanation: This is the power rule for integration.
19
Evaluate .
Correct Answer:
Explanation: The integral of the reciprocal function is the natural logarithm.
20
Calculate .
Correct Answer:
Explanation: Using the form . Here , so the result is .
21
To integrate , the standard result is:
Correct Answer:
Explanation: This is a standard integration formula involving inverse tangent.
22
Which trigonometric substitution is suitable for ?
Correct Answer:
Explanation: Substituting utilizes the identity to simplify the root.
23
Which property allows splitting into ?
A. Linearity property
B. Additivity of limits
C. Change of variable
D. Integration by parts
Correct Answer: Additivity of limits
Explanation: The interval can be split at a point such that .
24
Evaluate .
Correct Answer:
Explanation: Using Wallis formula or property , . , so .
25
Find .
Correct Answer:
Explanation: This is the standard Quotient Rule formula.
26
What is the derivative of ?
Correct Answer:
Explanation: The standard derivative of is .
27
If , what is ?
Correct Answer:
Explanation: . Using power rule: .
28
Find if and .
Correct Answer:
Explanation: .
29
What is the form of partial fraction for ?
A.
B.
C.
D. None of these
Correct Answer:
Explanation: Since all factors in the denominator are linear and distinct, we use constants over each factor.
30
Evaluate .
Correct Answer:
Explanation: Using integration by parts with , we get .
31
If , find the value of .
Correct Answer:
Explanation: Let be the integral. Using the property, . Adding the two forms, . Thus .
32
Find the derivative of .
Correct Answer:
Explanation: Using chain rule and exponential rule: . Here , so .
33
Determine the partial fraction decomposition form for .
Correct Answer:
Explanation: is an irreducible quadratic factor, so its numerator must be linear ().
34
Calculate .
Correct Answer:
Explanation: .
35
What is ?
Correct Answer:
Explanation: The derivative of the hyperbolic sine function is the hyperbolic cosine function.
36
If , differentiate to find at the point where derivatives exist.
Correct Answer:
Explanation: Implicit differentiation: . Solving for gives .
37
Evaluate .
Correct Answer:
Explanation: .
38
Which rule is applied to differentiate ?
A. Product Rule
B. Chain Rule
C. Quotient Rule
D. Power Rule
Correct Answer: Quotient Rule
Explanation: Since is a ratio of two functions, the Quotient Rule is applicable.
39
Evaluate .
Correct Answer:
Explanation: This is the standard integral resulting in the inverse sine function. Note there is no factor outside.
40
If , then is:
Correct Answer:
Explanation: Reversing the limits of a definite integral changes the sign of the result.
41
Find .
Correct Answer:
Explanation: According to the Fundamental Theorem of Calculus (Part 1), the derivative of is .
42
In integration by parts for , the best choice for is:
Correct Answer:
Explanation: Using ILATE, Algebraic () comes before Exponential (), so .
43
If , what is the first step in finding the derivative?
A. Use the power rule
B. Take the natural logarithm of both sides
C. Use the product rule immediately
D. Integrate both sides
Correct Answer: Take the natural logarithm of both sides
Explanation: Since the function is of the form , logarithmic differentiation is required.
44
Evaluate .
Correct Answer:
Explanation: The integral is .
45
What is the derivative of a constant function ?
Correct Answer: $0$
Explanation: The rate of change of a constant value is zero.
46
What is the partial fraction form for ?
Correct Answer:
Explanation: For repeated linear factors, we include terms with increasing powers of the factor.
47
Which of the following represents the Chain Rule?
Correct Answer:
Explanation: The chain rule states the derivative of a composite function is the product of the derivatives.
48
Calculate .
Correct Answer:
Explanation: Since , the integral of is .
49
If is an even function, what is ?
Correct Answer:
Explanation: For an even function (symmetric about y-axis), the area from to $0$ equals the area from $0$ to .
50
Find for .
Correct Answer:
Explanation: . The derivative is .