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Question 1
The derivative of a function $f$ is given by $f^{'} (x) = 6 e^{- 3 x}$ and $f (0) = 2$ . Find $f (x)$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $f (x) = - 2 e^{- 3 x} + 4$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Set up the integration
We need $f (x) = \int 6 e^{- 3 x} d x$ . Using the rule $\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + C$ with $a = - 3$ .
Step 2: Integrate
$f (x) = 6 \cdot \frac{1}{- 3} e^{- 3 x} + C = - 2 e^{- 3 x} + C$
Step 3: Use the initial condition $f(0) = 2$
Substitute $x = 0$ : $- 2 e^{0} + C = 2 \Rightarrow - 2 + C = 2 \Rightarrow C = 4$ .
Step 4: Write the final answer
$f (x) = - 2 e^{- 3 x} + 4$
Method #2Approach 2
Step 1: Identify what is needed
We need the antiderivative of $6 e^{- 3 x}$ that satisfies $f (0) = 2$ . Check each option by differentiating and applying $f (0) = 2$ .
Step 2: Eliminate $-2e^{-3x} + 2$
Differentiating $- 2 e^{- 3 x} + 2$ gives $6 e^{- 3 x}$ ✓, but $f (0) = - 2 + 2 = 0 \neq = 2$ . Eliminated.
Step 3: Eliminate $2e^{-3x} + 4$
Differentiating $2 e^{- 3 x} + 4$ gives $- 6 e^{- 3 x} \neq = 6 e^{- 3 x}$ . Eliminated.
Step 4: Eliminate $-18e^{-3x} + 20$
Differentiating $- 18 e^{- 3 x} + 20$ gives $54 e^{- 3 x} \neq = 6 e^{- 3 x}$ . Eliminated.
Step 5: Select the correct answer
The only remaining option is $- 2 e^{- 3 x} + 4$ . Check: derivative is $6 e^{- 3 x}$ ✓ and $f (0) = - 2 + 4 = 2$ ✓.
Question 2
Find $\int cos (5 x - 2) d x$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
C $\frac{1}{5} sin (5 x - 2) + C$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the form
The integrand is $cos (a x + b)$ with $a = 5$ and $b = - 2$ . Use the rule $\int cos (a x + b) d x = \frac{1}{a} sin (a x + b) + C$ .
Step 2: Apply the formula
$\int cos (5 x - 2) d x = \frac{1}{5} sin (5 x - 2) + C$
Step 3: Verify by differentiation
Differentiate $\frac{1}{5} sin (5 x - 2)$ : by the chain rule, $\frac{1}{5} \cdot cos (5 x - 2) \cdot 5 = cos (5 x - 2)$ ✓.
Step 4: State the answer
The correct answer is $\frac{1}{5} sin (5 x - 2) + C$ .
Method #2Approach 2
Step 1: What are we looking for?
The antiderivative of $cos (5 x - 2)$ . The antiderivative of $cos$ is $sin$ , and we divide by the inner derivative $a = 5$ .
Step 2: Eliminate $-\frac{1}{5}\sin(5x-2)+C$
This has a negative sign, which would correspond to integrating $- cos$ , not $+ cos$ . Eliminated.
Step 3: Eliminate $5\sin(5x-2)+C$
Differentiating gives $5 \cdot 5 cos (5 x - 2) = 25 cos (5 x - 2) \neq = cos (5 x - 2)$ . The constant is wrong. Eliminated.
Step 4: Eliminate $-5\cos(5x-2)+C$
This is a $cos$ result, but differentiating $cos$ gives $- sin$ , not $cos$ . The function type is wrong. Eliminated.
Step 5: Select the correct answer
$\frac{1}{5} sin (5 x - 2) + C$ is correct. Differentiation confirms: $\frac{1}{5} \cdot 5 cos (5 x - 2) = cos (5 x - 2)$ ✓.

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13 more questions in this topic

Free preview 2/15

Question 1
The derivative of a function $f$ is given by $f^{'} (x) = 6 e^{- 3 x}$ and $f (0) = 2$ . Find $f (x)$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $f (x) = - 2 e^{- 3 x} + 4$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Set up the integration
We need $f (x) = \int 6 e^{- 3 x} d x$ . Using the rule $\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + C$ with $a = - 3$ .
Step 2: Integrate
$f (x) = 6 \cdot \frac{1}{- 3} e^{- 3 x} + C = - 2 e^{- 3 x} + C$
Step 3: Use the initial condition $f(0) = 2$
Substitute $x = 0$ : $- 2 e^{0} + C = 2 \Rightarrow - 2 + C = 2 \Rightarrow C = 4$ .
Step 4: Write the final answer
$f (x) = - 2 e^{- 3 x} + 4$
Method #2Approach 2
Step 1: Identify what is needed
We need the antiderivative of $6 e^{- 3 x}$ that satisfies $f (0) = 2$ . Check each option by differentiating and applying $f (0) = 2$ .
Step 2: Eliminate $-2e^{-3x} + 2$
Differentiating $- 2 e^{- 3 x} + 2$ gives $6 e^{- 3 x}$ ✓, but $f (0) = - 2 + 2 = 0 \neq = 2$ . Eliminated.
Step 3: Eliminate $2e^{-3x} + 4$
Differentiating $2 e^{- 3 x} + 4$ gives $- 6 e^{- 3 x} \neq = 6 e^{- 3 x}$ . Eliminated.
Step 4: Eliminate $-18e^{-3x} + 20$
Differentiating $- 18 e^{- 3 x} + 20$ gives $54 e^{- 3 x} \neq = 6 e^{- 3 x}$ . Eliminated.
Step 5: Select the correct answer
The only remaining option is $- 2 e^{- 3 x} + 4$ . Check: derivative is $6 e^{- 3 x}$ ✓ and $f (0) = - 2 + 4 = 2$ ✓.
Question 2
Find $\int cos (5 x - 2) d x$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
C $\frac{1}{5} sin (5 x - 2) + C$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the form
The integrand is $cos (a x + b)$ with $a = 5$ and $b = - 2$ . Use the rule $\int cos (a x + b) d x = \frac{1}{a} sin (a x + b) + C$ .
Step 2: Apply the formula
$\int cos (5 x - 2) d x = \frac{1}{5} sin (5 x - 2) + C$
Step 3: Verify by differentiation
Differentiate $\frac{1}{5} sin (5 x - 2)$ : by the chain rule, $\frac{1}{5} \cdot cos (5 x - 2) \cdot 5 = cos (5 x - 2)$ ✓.
Step 4: State the answer
The correct answer is $\frac{1}{5} sin (5 x - 2) + C$ .
Method #2Approach 2
Step 1: What are we looking for?
The antiderivative of $cos (5 x - 2)$ . The antiderivative of $cos$ is $sin$ , and we divide by the inner derivative $a = 5$ .
Step 2: Eliminate $-\frac{1}{5}\sin(5x-2)+C$
This has a negative sign, which would correspond to integrating $- cos$ , not $+ cos$ . Eliminated.
Step 3: Eliminate $5\sin(5x-2)+C$
Differentiating gives $5 \cdot 5 cos (5 x - 2) = 25 cos (5 x - 2) \neq = cos (5 x - 2)$ . The constant is wrong. Eliminated.
Step 4: Eliminate $-5\cos(5x-2)+C$
This is a $cos$ result, but differentiating $cos$ gives $- sin$ , not $cos$ . The function type is wrong. Eliminated.
Step 5: Select the correct answer
$\frac{1}{5} sin (5 x - 2) + C$ is correct. Differentiation confirms: $\frac{1}{5} \cdot 5 cos (5 x - 2) = cos (5 x - 2)$ ✓.

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Step-by-step walkthrough

Step 1: Set up the integration

Step 2: Integrate

Step 3: Use the initial condition $f(0) = 2$

Step 4: Write the final answer

Step 1: Identify what is needed

Step 2: Eliminate $-2e^{-3x} + 2$

Step 3: Eliminate $2e^{-3x} + 4$

Step 4: Eliminate $-18e^{-3x} + 20$

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the form

Step 2: Apply the formula

Step 3: Verify by differentiation

Step 4: State the answer

Step 1: What are we looking for?

Step 2: Eliminate $-\frac{1}{5}\sin(5x-2)+C$

Step 3: Eliminate $5\sin(5x-2)+C$

Step 4: Eliminate $-5\cos(5x-2)+C$

Step 5: Select the correct answer

13 more questions in this topic

Step-by-step walkthrough

Step 1: Set up the integration

Step 2: Integrate

Step 3: Use the initial condition $f(0) = 2$

Step 4: Write the final answer

Step 1: Identify what is needed

Step 2: Eliminate $-2e^{-3x} + 2$

Step 3: Eliminate $2e^{-3x} + 4$

Step 4: Eliminate $-18e^{-3x} + 20$

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the form

Step 2: Apply the formula

Step 3: Verify by differentiation

Step 4: State the answer

Step 1: What are we looking for?

Step 2: Eliminate $-\frac{1}{5}\sin(5x-2)+C$

Step 3: Eliminate $5\sin(5x-2)+C$

Step 4: Eliminate $-5\cos(5x-2)+C$

Step 5: Select the correct answer

13 more questions in this topic