The temperature model is . What is ?
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AP Calculus BC Quiz
Practice Applying The Power Rule in AP Calculus BC with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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The temperature model is T(t)=t8−6t4+9t2+1. What is T′(t)?
This quiz focuses on Applying The Power Rule, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus BC.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
The temperature model is T(t)=t8−6t4+9t2+1. What is T′(t)?
Explanation: This temperature model requires applying the power rule to find the rate of change. The power rule states that for any term at^n, the derivative is n·at^(n-1). For T(t) = t^8 - 6t^4 + 9t^2 + 1, we differentiate each term: t^8 becomes 8t^7, -6t^4 becomes -24t^3, 9t^2 becomes 18t, and the constant 1 becomes 0. Therefore, T'(t) = 8t^7 - 24t^3 + 18t. Choice D incorrectly adds the constant 1 to the derivative, failing to recognize that constants differentiate to zero. Remember the power rule strategy: bring down the exponent as a multiplier, reduce the exponent by 1, and constants vanish.
A particle’s position is modeled by s(t)=4t7−3t4+6t−9. What is s′(t)?
Explanation: This problem requires applying the power rule to find the derivative of a position function. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). For s(t) = 4t^7 - 3t^4 + 6t - 9, we apply the rule term by term: the derivative of 4t^7 is 28t^6, the derivative of -3t^4 is -12t^3, the derivative of 6t is 6, and the derivative of the constant -9 is 0. Choice E incorrectly keeps the constant term -9, which is a common error since constants have zero derivatives. Remember: when using the power rule, multiply the coefficient by the exponent and reduce the exponent by 1.
A cost function is C(x)=x9−4x5+2x2−1. What is C′(x)?
Explanation: Finding C'(x) requires applying the power rule to the cost function. The power rule states that the derivative of x^n is nx^(n-1). For C(x) = x^9 - 4x^5 + 2x^2 - 1, we differentiate each term: x^9 becomes 9x^8, -4x^5 becomes -20x^4, 2x^2 becomes 4x, and the constant -1 becomes 0. Choice A incorrectly keeps the constant -1 in the derivative, which violates the rule that constants have zero derivatives. Always drop constant terms when applying the power rule to find derivatives.
A polynomial model for revenue is R(x)=x7−4x5+6x2. What is R′(x)?
Explanation: The power rule is the key skill here, which states that the derivative of a term like a x^n is a n x^{n-1}. To find R'(x) from R(x) = x^7 - 4x^5 + 6x^2, begin with the first term: the derivative of x^7 (which is 1x^7) is 17 x^6 = 7x^6. Then, the derivative of -4x^5 is -45 x^4 = -20x^4, and the derivative of 6x^2 is 6*2 x^1 = 12x. Combining these gives 7x^6 - 20x^4 + 12x. A tempting distractor like choice D, 7x^6 - 4x^4 + 12x, fails by forgetting to multiply the -4 by the exponent 5. Remember, a transferable strategy for the power rule is to handle each term independently, multiply by the exponent, and subtract one from the exponent, ensuring constants vanish.
A cost function is C(x)=9x4−2x3+x−12. What is C′(x)?
Explanation: The power rule is the key skill here, which states that the derivative of a term like a x^n is a n x^{n-1}, and constants have a derivative of zero. To find C'(x) from C(x) = 9x^4 - 2x^3 + x - 12, begin with the first term: the derivative of 9x^4 is 94 x^3 = 36x^3. Then, the derivative of -2x^3 is -23 x^2 = -6x^2, and the derivative of x (which is 1x^1) is 1*1 x^0 = 1. The constant -12 differentiates to 0, resulting in 36x^3 - 6x^2 + 1. A tempting distractor like choice D, 36x^3 - 2x^2 + 1, fails by not multiplying the -2 coefficient by its exponent of 3. Remember, a transferable strategy for the power rule is to handle each term independently, multiply by the exponent, and subtract one from the exponent, ensuring constants vanish.
The displacement is d(x)=−6x5+11x2−4x+13. What is d′(x)?
Explanation: To find d'(x) for the displacement function d(x) = -6x^5 + 11x^2 - 4x + 13, we apply the power rule to each term. Using the rule that the derivative of x^n is nx^(n-1): the derivative of -6x^5 is 5·(-6)x^4 = -30x^4, the derivative of 11x^2 is 2·11x^1 = 22x, the derivative of -4x is -4·1x^0 = -4, and the derivative of 13 is 0. Choice C (-6x^4 + 11x - 4) makes the error of only reducing exponents without multiplying by them, missing the key multiplication step in the power rule. Remember: multiply the coefficient by the exponent, then reduce the exponent by 1.
A firm’s revenue is R(x)=3x7+4x2−x+8. What is R′(x)?
Explanation: Finding R'(x) for the revenue function R(x) = 3x^7 + 4x^2 - x + 8 requires applying the power rule to each term. The power rule gives us: the derivative of 3x^7 is 7·3x^6 = 21x^6, the derivative of 4x^2 is 2·4x^1 = 8x, the derivative of -x (which is -x^1) is 1·(-1)x^0 = -1, and the derivative of 8 is 0. Choice C (3x^6 + 4x - 1) incorrectly applies the power rule by not multiplying the coefficients by the exponents for the first two terms. Remember that the power rule requires both multiplication by the exponent and reduction of the exponent by 1.
The cost function is C(x)=12x4−5x3+2x+11. What is C′(x)?
Explanation: To find C'(x), we apply the power rule to each term of the cost function C(x) = 12x^4 - 5x^3 + 2x + 11. Using the power rule d/dx(x^n) = nx^(n-1), we get: the derivative of 12x^4 is 4·12x^3 = 48x^3, the derivative of -5x^3 is 3·(-5)x^2 = -15x^2, the derivative of 2x is 2·1x^0 = 2, and the derivative of the constant 11 is 0. Choice C (12x^3 - 5x^2 + 2) makes the common error of only bringing down the exponent without multiplying by the original coefficient. The key to the power rule is to multiply the coefficient by the exponent before reducing the exponent by 1.
A ball’s height is modeled by h(t)=5t4−3t2+7t−9 meters; what is h′(t)?
Explanation: This problem involves using the power rule to differentiate the polynomial h(t)=5t4−3t2+7t−9. To apply the power rule, take each term separately: for 5t4, the derivative is 5∗4t3=20t3. Next, for −3t2, it's −3∗2t1=−6t. Then, for 7t, it's 7∗1t0=7, and the constant −9 differentiates to 0. A common mistake is seen in choice C, where exponents aren't reduced, leading to incorrect terms like 20t4 instead of decreasing the powers. Remember, a key strategy for the power rule is to multiply each term's coefficient by its exponent and then subtract one from that exponent, ensuring constants disappear in the derivative.
A temperature model is T(t)=3t−2−5t4+11; which expression equals T′(t)?
Explanation: This problem involves the power rule for T(t) = 3t^{-2} - 5t^4 + 11, featuring a negative exponent. Differentiate 3t^{-2} to 3*(-2) t^{-3} = -6t^{-3}. For -5t^4, it's -5*4 t^3 = -20t^3. The constant +11 becomes 0. Choice D is tempting but wrong as it omits the negative sign from the first term's differentiation. A general strategy for the power rule, even with negative or fractional exponents, is to multiply each coefficient by its exponent and subtract one from the exponent, dropping constants.
A function is defined by f(x)=x−4+3x−2−2x. What is f′(x)?
Explanation: The power rule is the key skill here, which states that the derivative of a term like a x^n is a n x^{n-1}, even for negative exponents. To find f'(x) from f(x) = x^{-4} + 3 x^{-2} - 2x, start with the first term: the derivative of x^{-4} is 1*(-4) x^{-5} = -4 x^{-5}. Next, the derivative of 3 x^{-2} is 3*(-2) x^{-3} = -6 x^{-3}, and the derivative of -2x is -2*1 x^0 = -2. This combines to -4 x^{-5} - 6 x^{-3} - 2. A tempting distractor like choice E, -4 x^{-5} - 6 x^{-3} - 2x, fails because it incorrectly treats the last term as having an exponent that doesn't reduce to 0. Remember, a transferable strategy for the power rule is to handle each term independently, multiply by the exponent, and subtract one from the exponent, ensuring constants vanish.
A cost function is C(q)=9q7+q2−6q+1; what is the marginal cost C′(q)?
Explanation: This problem uses the power rule to find the marginal cost from C(q) = 9q^7 + q^2 - 6q + 1. Differentiate 9q^7 to 97 q^6 = 63q^6. For q^2, it's 12 q^1 = 2q. Then, -6q becomes -6*1 q^0 = -6, and +1 is 0. Choice C fails by not reducing exponents, keeping q^7. A key strategy is to apply the power rule term by term, multiplying by the exponent and subtracting one, with constants dropping out of the derivative.
The temperature model is T(t)=3t8−2t5+11t2. What is T′(t)?
Explanation: To find T'(t), we apply the power rule to differentiate the temperature function. The power rule states that if f(t) = t^n, then f'(t) = nt^(n-1). For T(t) = 3t^8 - 2t^5 + 11t^2, we get: the derivative of 3t^8 is 8·3t^7 = 24t^7, the derivative of -2t^5 is 5·(-2)t^4 = -10t^4, and the derivative of 11t^2 is 2·11t = 22t. Choice C incorrectly keeps the original coefficients without multiplying by the exponents, showing incomplete understanding of the power rule. The complete power rule process: multiply coefficient by exponent, then subtract 1 from the exponent.
A sensor output is f(x)=x7−4x4+2x2−11. What is f′(x)?
Explanation: This sensor output function requires applying the power rule to find its derivative. The power rule states that for any term ax^n, the derivative is n·ax^(n-1). For f(x) = x^7 - 4x^4 + 2x^2 - 11, we differentiate each term: x^7 becomes 7x^6, -4x^4 becomes -16x^3, 2x^2 becomes 4x, and the constant -11 becomes 0. Therefore, f'(x) = 7x^6 - 16x^3 + 4x. Choice D incorrectly includes the original constant -11 in the derivative, not recognizing that constants differentiate to zero. The power rule strategy is straightforward: for each term, multiply the coefficient by the exponent and subtract 1 from the exponent.
The revenue model is R(x)=9x4−2x3+6. What is R′(x)?
Explanation: This problem asks us to find the derivative of a revenue function using the power rule. The power rule tells us that the derivative of x^n is nx^(n-1). For R(x) = 9x^4 - 2x^3 + 6, we differentiate term by term: the derivative of 9x^4 is 4·9x^3 = 36x^3, the derivative of -2x^3 is 3·(-2)x^2 = -6x^2, and the derivative of the constant 6 is 0. Thus, R'(x) = 36x^3 - 6x^2. Choice D incorrectly adds the constant 6 to the derivative, failing to recognize that constants have zero derivatives. The key strategy with the power rule is to process each term independently: multiply by the exponent, then decrease the exponent by 1.
A particle’s position is s(t)=4t5−3t2+7t−9. What is s′(t)?
Explanation: This problem requires applying the power rule to find the derivative of a polynomial position function. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). For s(t) = 4t^5 - 3t^2 + 7t - 9, we apply the power rule to each term: the derivative of 4t^5 is 5·4t^4 = 20t^4, the derivative of -3t^2 is 2·(-3)t^1 = -6t, the derivative of 7t is 7·1t^0 = 7, and the derivative of the constant -9 is 0. Therefore, s'(t) = 20t^4 - 6t + 7. Choice E incorrectly keeps the linear term 7t in addition to its derivative 7, showing a common error of not replacing terms with their derivatives. Remember: when applying the power rule, multiply the coefficient by the exponent and reduce the exponent by 1 for each term.
A polynomial is m(x)=3x10−x5+4x3+9. What is m′(x)?
Explanation: To find m'(x), we apply the power rule to each term of the polynomial. The power rule gives us that the derivative of ax^n is nax^(n-1). For m(x) = 3x^10 - x^5 + 4x^3 + 9, we differentiate term by term: 3x^10 becomes 30x^9, -x^5 becomes -5x^4, 4x^3 becomes 12x^2, and the constant 9 becomes 0. Choice E incorrectly includes the constant 9 in the derivative, which is wrong since constants differentiate to zero. Remember the power rule process: multiply coefficient by exponent, reduce exponent by 1, and eliminate constants.
A polynomial model is f(x)=−6x7+3x4−x3+10. What is f′(x)?
Explanation: To find f'(x), we apply the power rule to each term of the polynomial. The power rule tells us that d/dx[ax^n] = nax^(n-1). For f(x) = -6x^7 + 3x^4 - x^3 + 10, we differentiate term by term: -6x^7 becomes -42x^6, 3x^4 becomes 12x^3, -x^3 becomes -3x^2, and the constant 10 becomes 0. Choice B incorrectly includes the constant 10, failing to recognize that constants differentiate to zero. Remember the power rule pattern: bring down the exponent as a multiplier, then subtract 1 from the exponent.
The height of water is approximated by h(t)=2t8−7t2+3. What is h′(t)?
Explanation: To find h'(t), we apply the power rule to each term in the height function. The power rule gives us that if f(x) = ax^n, then f'(x) = nax^(n-1). For h(t) = 2t^8 - 7t^2 + 3, we differentiate term by term: 2t^8 becomes 16t^7, -7t^2 becomes -14t, and the constant 3 becomes 0. Choice A incorrectly includes the constant 3 in the derivative, which is a mistake since constants differentiate to zero. When using the power rule, remember to eliminate all constant terms from your derivative.
The revenue function is R(x)=5x6−2x3+11. What is R′(x)?
Explanation: To find R'(x), we need to apply the power rule to each term of the revenue function. The power rule tells us that the derivative of x^n is nx^(n-1). For R(x) = 5x^6 - 2x^3 + 11, we differentiate term by term: 5x^6 becomes 30x^5, -2x^3 becomes -6x^2, and the constant 11 becomes 0. Choice A incorrectly includes the constant 11 in the derivative, which is wrong because the derivative of any constant is zero. When applying the power rule, always remember to drop constant terms from your final answer.