Calculus 12 Final Exam Review (Part 2)

Riemann Sums

100

What variable represents the width of a rectangle/sub-interval when using Riemann sums to estimate the integral of a function? How is it calculated? 

Width = Δx. It is calculated by taking the difference between the limits of integration and dividing it by the number of sub-intervals. 

200

Estimate the area under the curve of the function y = 7x² on the interval [0,2] using a right hand approximation with 4 sub-intervals.

105/4 or 26.25

300

Use a left hand approximation with 3 sub-intervals to estimate the area under the curve of the function y = 2/x on the interval [0.5, 2]. 

11/3 

400

Use the midpoint rule with 5 sub-intervals to estimate the integral of 1/x dx on the interval [1,2]. 

0.69

500

Use the midpoint rule to approximate the integral sin(sqrt(x)) dx with four sub-intervals. 

6.182

Integrals

100

Find the definite integral of 12x² + 1 dx on the interval [-2,7]. 

1413

200

Find the definite integral of x² + 2x - 2 dx on the interval [-3,0]. 

-6

300

Find the indefinite integral of 12 dy. 

12y + C

400

Find the indefinite integral of (4)/(x²) + 2 - (1)/(8x³) dx.

-4x^-1 + 2x + (1/16)x^-2 + C

500

Find the indefinite integral of (x² - 1)(4 + 3x) dx.

(3/4)x⁴ + (4/3)x³ - (3/2)x² - 4x + C

Differential Equations

100

What is the general solution of the differential equation dy/dx = -1/x²?

y = (1)/(x) + C

200

What is the general solution of the differential equation dy/dx = xe^y?

y = -ln(-(x²)/(2) + C)

300

Based on the initial condition y(1) = 4, find the particular solution of the differential equation dy/dx = 1/x². 

y = -(1)/(x) + 5

400

Based on the initial condition y(0) = 4, find the particular solution of the differential equation dy/dx = 3x² - 1. 

x³ - x + 4

500

Based on the initial condition y(0) = -3, find the particular solution of the differential equation dy/dx = 2xy. 

y = -3e^{x^2}

Exponential Growth and Decay

100

What does each variable represent in the general formula for exponential growth and decay? 

y = final value 

y_o = initial value 

k = continuous growth/decay rate or constant of proportionality 

t = time

200

What is the general formula for exponential growth and decay?

y = y_oe^kt

300

A bacteria population starts with 4000 bacteria. After 3 hours, the population had increased to 7000. What is the value of k? 

ln(7/4) / 3 or 0.19

400

The rabbit population on a certain island was estimated to be 1500 in the year 2000. In the year 2001, it was estimated to be 1577. What is the relative growth rate of the population (k)? 

k = 0.05 

500

Based on the 400-point question for Exponential Growth and Decay, estimate the rabbit population in 2010. 

2473

General Knowledge

100

When was calculus invented? 

In the late 17th century 

200

Who invented calculus? (two people)

Isaac Newton and Gottfried Leibniz 

300

What are the two main branches of calculus?

Differential and Integral 

400

What is a limit?

The value that y approaches as x gets closer and closer to a given number. 

500

What is the derivative?

The instantaneous rate of change of a function or the slope of the tangent line of a function at a certain point.