Math Quiz: Take The Limits And Continuity Assessment Test

1. A function f(x) passes through the origin and its first derivative is 3x + 2. What is f(x)?

Y=3/2x2 + 2x

Y=3/2x2 + x

Y=3x2 + x/2

Y=3x2 + 2x

The given information states that the function passes through the origin, which means that the y-intercept is 0. The first derivative of the function is given as 3x + 2. To find the original function, we need to integrate the first derivative. Integrating 3x + 2 gives us (3/2)x^2 + 2x + C, where C is the constant of integration. Since the function passes through the origin, the constant of integration is 0. Therefore, the original function is y = (3/2)x^2 + 2x.

Explanation

The given information states that the function passes through the origin, which means that the y-intercept is 0. The first derivative of the function is given as 3x + 2. To find the original function, we need to integrate the first derivative. Integrating 3x + 2 gives us (3/2)x^2 + 2x + C, where C is the constant of integration. Since the function passes through the origin, the constant of integration is 0. Therefore, the original function is y = (3/2)x^2 + 2x.

2. Evaluate the limit ( x – 4 ) / (x2 – x – 12) as x approaches 4.

0

Infinity

1/7

Undefined

As x approaches 4, we can substitute the value of x into the expression. This gives us (4 - 4) / (4^2 - 4 - 12) = 0 / (16 - 4 - 12) = 0 / 0. However, dividing by 0 is undefined. Therefore, the limit is undefined.

Explanation

As x approaches 4, we can substitute the value of x into the expression. This gives us (4 - 4) / (4^2 - 4 - 12) = 0 / (16 - 4 - 12) = 0 / 0. However, dividing by 0 is undefined. Therefore, the limit is undefined.

3. Differentiate y = sec (x2 + 2).

2x cos (x2 + 2)

–cos (x2 + 2) cot (x2 + 2)

2x sec (x2 + 2) tan (x2 + 2)

Cos (x2 +2)

The given question asks for the differentiation of the function y = sec(x^2 + 2). The correct answer is 2x sec(x^2 + 2) tan(x^2 + 2). This can be obtained by applying the chain rule of differentiation. The derivative of sec(x) is sec(x) tan(x), and since the function inside the sec function is x^2 + 2, we need to multiply the derivative of x^2 + 2, which is 2x, with sec(x^2 + 2) tan(x^2 + 2). Therefore, the correct answer is 2x sec(x^2 + 2) tan(x^2 + 2).

Explanation

The given question asks for the differentiation of the function y = sec(x^2 + 2). The correct answer is 2x sec(x^2 + 2) tan(x^2 + 2). This can be obtained by applying the chain rule of differentiation. The derivative of sec(x) is sec(x) tan(x), and since the function inside the sec function is x^2 + 2, we need to multiply the derivative of x^2 + 2, which is 2x, with sec(x^2 + 2) tan(x^2 + 2). Therefore, the correct answer is 2x sec(x^2 + 2) tan(x^2 + 2).

4. Differentiate the equation y = x2 / (x +1).

2x

(x2 + 2x) / (x + 1)2

X / (x + 1)

(2×2) / (x + 1)

The given equation is y = x^2 / (x + 1). To differentiate this equation, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2. Applying this rule to the given equation, we have g(x) = x^2 and h(x) = (x + 1). Taking the derivatives of g(x) and h(x), we get g'(x) = 2x and h'(x) = 1. Plugging these values into the quotient rule formula, we get the differentiated equation as (x^2 + 2x) / (x + 1)^2.

Explanation

The given equation is y = x^2 / (x + 1). To differentiate this equation, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2. Applying this rule to the given equation, we have g(x) = x^2 and h(x) = (x + 1). Taking the derivatives of g(x) and h(x), we get g'(x) = 2x and h'(x) = 1. Plugging these values into the quotient rule formula, we get the differentiated equation as (x^2 + 2x) / (x + 1)^2.

5. If y=xsinx, find dx/dy.

Sinx - cosx

Cosx - xsinx

Cosx + xsinx

Sinx + xcox

The correct answer is sinx + xcox. To find dx/dy, we need to differentiate y with respect to x and then take the reciprocal. Differentiating y = xsinx using the product rule, we get dy/dx = sinx + xcox. Taking the reciprocal, we get dx/dy = 1 / (sinx + xcox).

Explanation

The correct answer is sinx + xcox. To find dx/dy, we need to differentiate y with respect to x and then take the reciprocal. Differentiating y = xsinx using the product rule, we get dy/dx = sinx + xcox. Taking the reciprocal, we get dx/dy = 1 / (sinx + xcox).

6. Find dy/dx if y = 52x-1.

52x-1 ln 5

52x-1 ln 25

52x-1 ln 10

52x-1 ln 2

The correct answer is 52x-1 ln 25. To find the derivative of y with respect to x, we use the power rule of differentiation. The power rule states that if y = ax^n, then dy/dx = nax^(n-1). In this case, n = -1 and a = 52. Taking the derivative, we get dy/dx = -52x^(-1-1) = -52/x^2. Since ln(25) is a constant, it does not affect the derivative. Therefore, the correct answer is 52x-1 ln 25.

Explanation

The correct answer is 52x-1 ln 25. To find the derivative of y with respect to x, we use the power rule of differentiation. The power rule states that if y = ax^n, then dy/dx = nax^(n-1). In this case, n = -1 and a = 52. Taking the derivative, we get dy/dx = -52x^(-1-1) = -52/x^2. Since ln(25) is a constant, it does not affect the derivative. Therefore, the correct answer is 52x-1 ln 25.

7. Evaluate the limit (1n x ) / x as x approaches positive infinity.

1

0

E

Infinity

As x approaches positive infinity, the numerator (ln x) approaches infinity because the natural logarithm of any positive number grows without bound as the number increases. However, the denominator (x) also approaches infinity. As a result, the fraction (ln x) / x approaches 0, since the numerator grows faster than the denominator. Therefore, the limit of the expression is 0.

Explanation

As x approaches positive infinity, the numerator (ln x) approaches infinity because the natural logarithm of any positive number grows without bound as the number increases. However, the denominator (x) also approaches infinity. As a result, the fraction (ln x) / x approaches 0, since the numerator grows faster than the denominator. Therefore, the limit of the expression is 0.

8. Differentiate (x2 + 2)1/2.

(x2 + 2)1/2) / 2

X / (x2 + 2)1/2

(2x) / (x2 + 2)1/2

(x2 + 2)3/2

The given options are all derivatives of the expression (x^2 + 2)^(1/2). However, the correct answer is x / (x^2 + 2)^(1/2). This can be determined by using the power rule for differentiation. The power rule states that if we have a function of the form (f(x))^n, then its derivative is n * (f(x))^(n-1) * f'(x). In this case, f(x) = x^2 + 2 and n = 1/2. Taking the derivative, we get 1/2 * (x^2 + 2)^(-1/2) * (2x) = x / (x^2 + 2)^(1/2).

Explanation

The given options are all derivatives of the expression (x^2 + 2)^(1/2). However, the correct answer is x / (x^2 + 2)^(1/2). This can be determined by using the power rule for differentiation. The power rule states that if we have a function of the form (f(x))^n, then its derivative is n * (f(x))^(n-1) * f'(x). In this case, f(x) = x^2 + 2 and n = 1/2. Taking the derivative, we get 1/2 * (x^2 + 2)^(-1/2) * (2x) = x / (x^2 + 2)^(1/2).

9. Find the point in the parabola y2 = 4x at which the rate of change of the ordinate and abscissa are equal.

(4,4)

(2, 1)

(1,2)

(-1, 4)

The point (1,2) is the correct answer because it satisfies the condition that the rate of change of the ordinate (y-coordinate) and abscissa (x-coordinate) are equal. In the given parabola equation y^2 = 4x, we can find the rate of change of y with respect to x by differentiating the equation with respect to x. The derivative of y^2 with respect to x is 2y(dy/dx) = 4. Simplifying this equation, we get dy/dx = 2/y. At the point (1,2), the y-coordinate is 2, so the rate of change of y with respect to x is 2/2 = 1. Therefore, the rate of change of the ordinate and abscissa are equal at the point (1,2).

Explanation

The point (1,2) is the correct answer because it satisfies the condition that the rate of change of the ordinate (y-coordinate) and abscissa (x-coordinate) are equal. In the given parabola equation y^2 = 4x, we can find the rate of change of y with respect to x by differentiating the equation with respect to x. The derivative of y^2 with respect to x is 2y(dy/dx) = 4. Simplifying this equation, we get dy/dx = 2/y. At the point (1,2), the y-coordinate is 2, so the rate of change of y with respect to x is 2/2 = 1. Therefore, the rate of change of the ordinate and abscissa are equal at the point (1,2).

10. Locate the points of inflection of the curve y = f(x) = x2 ex.

-2 ± √2

2 ± √2

2 ± √3

The points of inflection of a curve occur where the concavity changes. To find these points, we need to find the second derivative of the curve. Taking the derivative of f(x) = x^2 * e^x, we get f''(x) = 2e^x + 2xe^x. Setting this equal to zero and solving for x, we find that x = -2 ± √2. Therefore, the points of inflection of the curve y = f(x) = x^2 * e^x are -2 ± √2.

Explanation

The points of inflection of a curve occur where the concavity changes. To find these points, we need to find the second derivative of the curve. Taking the derivative of f(x) = x^2 * e^x, we get f''(x) = 2e^x + 2xe^x. Setting this equal to zero and solving for x, we find that x = -2 ± √2. Therefore, the points of inflection of the curve y = f(x) = x^2 * e^x are -2 ± √2.