Free UTMA Calculus Readiness Questions and Answers

Question 1

What is the derivative of 𝑓(x)= x 3 ?

Accepted Answer

3x2

Answer

The derivative of f(x) = x^3 is found using the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this rule, the derivative of x^3 is 3 * x^(3-1). This simplifies to 3x^2.

Question 2

Evaluate the limit: lim𝑥→2 (3𝑥+1)

Accepted Answer

7

Answer

To evaluate the limit of a polynomial function as x approaches a specific value, simply substitute that value into the function. Substituting x=2 into the expression (3x+1) gives 3(2)+1. This calculation results in 6+1, which equals 7.

Question 3

What is the integral of 𝑓 (x)=2x?

Accepted Answer

𝑥2 + 𝐶

Answer

To find the integral of f(x) = 2x, we use the power rule for integration, which states that the integral of x^n is (x^(n+1))/(n+1). For 2x (or 2x^1), this becomes 2 * (x^(1+1))/(1+1). This simplifies to 2 * (x^2)/2 = x^2. Since it's an indefinite integral, we add the constant of integration, C, resulting in x^2 + C.

Question 4

What is the second derivative of 𝑓 (𝑥) = 𝑥4?

Accepted Answer

12x 2

Answer

To find the second derivative of f(x) = x^4, first calculate the first derivative using the power rule: f'(x) = 4x^3. Then, differentiate f'(x) again to find the second derivative: f''(x) = 4 * (3x^2). This simplifies to 12x^2.

Question 5

Find the critical points of 𝑓 (𝑥) =𝑥 3−3𝑥2

Accepted Answer

0 and 3

Answer

Critical points are found by taking the first derivative of a function and setting it to zero. If the derivative of a function factors into terms like 3x(x-3)=0, then solving for x would yield x=0 and x=3. These points indicate where the function's slope is horizontal, often corresponding to local maxima or minima.

UTMA - University of Toronto Math Assessment Practice Test

UTMA - University of Toronto Math Assessment Practice Test

Free UTMA Calculus Readiness Questions and Answers