General revision

Question 1

Find the derivative of f(x) = 3x^2 using first principles.

Answer 1

6x

Question 2

Using first principles, calculate the derivative of g(x) = (2x^3 + 5x^2 - 3x + 1).

Answer 2

6x^2 + 10x - 3.

Question 3

Compute the derivative of h(x) = 4x^5 - 2x^3 + 7x - 9.

Answer 3

20x^4 - 6x^2 + 7.

Question 4

Find the derivative of the function (3x^4 - 2x^3 + 5x^2 - 6x + 2).

Answer 4

(12x^3 - 6x^2 + 10x - 6)

Question 5

Apply the chain rule to find the derivative of j(x) = 12(2x^2+19)^2.

Answer 5

48x(2x^2+19)

Question 6

Differentiate k(x) = 5x^2 + 3x - 1

Answer 6

10x + 3

Question 7

Use the product rule to find the derivative of p(x) = x^2 * (3x + 1).

Answer 7

3x^3 + 2x^2.

Question 8

Apply the quotient rule to find the derivative of r(x) = (4x^2 - 1) / (2x + 3).

Answer 8

r ′ (x)=8x^2+24x+2) / (2x + 3)^2

Question 9

Determine the x-coordinate of the local maximum of the function f(x) = 2x^3 - 3x^2 + 6x - 1.

Answer 9

Lmao there isn’t one

Question 10

Find both x and y coordinates of all turning points of the function g(x) = x^4 - 4x^3 + 3x^2 + 2x + 5.

Answer 10

Just do it

Question 11

Calculate the x-values of the inflection points for h(x) = 3x^4 - 12x^3 + 6x^2 - 5x + 1.

Answer 11

Just do it

Question 12

Find the coordinates of the local minimum and maximum for i(x) = e^x * sin(x) in the interval [0, 2π].

Answer 12

Just do it

Question 13

Given the position function s(t) = 3t^2 - 2t + 1, find the velocity and acceleration functions.

Answer 13

Just do it

Question 14

A car’s velocity is given by v(t) = 6t^2 - 2t + 5. Find the car’s displacement and acceleration at time t = 3 seconds.

Answer 14

Just do it

Question 15

Sketch the graph of y = sin(x) for the interval [-2π, 2π].

Answer 15

Just do it

Question 16

Hard: Draw the graph of y = 3cos(2x+pi/2) -4for 1 period

Answer 16

Just do it

Question 17

Calculate the average rate of change of the function f(x) = 4x^2 - 2x + 3 over the interval [1, 3].

Answer 17

Just do it

Question 18

Determine the average rate of change of the function g(x) = ln(x^2 + 1) over the interval [0, 2].

Answer 18

Just do it

Question 19

Find the derivative of the function p(x) = (x^2 - 3x + 1)(2x^3 + 4x^2 - x).

Answer 19

p’(x) = 4x^4 - 3x^2 + 19x - 3.

Question 20

Find the derivative of the function q(x) = (x^4 + 2x^3 - x^2 + 5x - 3) / (2x^2 + 3x - 1).

Answer 20

q’(x) = (22x^2 - 6) / (2x^2 + 3x - 1)^2.

Question 21

Find the derivative of the following function: f(x) = [(x^2 + 1)(e^x)] / [(2x + 3)(ln(x^2 + 2x + 5))].

Answer 21

f’(x) = [(2xe^x + (x^2 + 1)e^x)(2ln(x^2 + 2x + 5)) - (2ln(x^2 + 2x + 5) + (2x + 3)(1 / (x^2 + 2x + 5))(2x + 2))(x^2 + 1)e^x] / [(2x + 3)(ln(x^2 + 2x + 5))]^2 Simplify and factor out common terms if possible. The final expression will be quite lengthy due to the complexity of the problem.