# High school calculus questions and answers Recent questions in Calculus and Analysis
Matrices

### Hello everyone! I'm a college student and taking a couse of Electrical Engineering. I hope you will help me with my homework. Thank you! The bases of a trapezoid are 22 and 12 respectively. The angles at the extremities of one base are 65° and 40° respectively. Find the two legs.

Applications of integrals

Matrices

### If the roots of $$ax^{2}+bx+c=0$$ are $$\frac{3}{2},\frac{4}{3}$$ then $$(a+b+c)^{2} =$$

Calculus and Analysis

### What is the best way of determining whether the given equation is a circle?

Derivatives
ANSWERED ### Determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$

Derivatives
ANSWERED ### Compute the gradient of the following functions and evaluate it at the given point P. $$\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{{2}}}-{4}{x}^{{{2}}}{y}-{8}{x}{y}^{{{2}}};{P}{\left(-{1},{2}\right)}$$

Derivatives
ANSWERED ### The radius of a sphere increases at a rate of $$\displaystyle{4}\frac{{m}}{{\sec{}}}$$. Find the rate at which the surface area increases when the radius is 9m. $$\displaystyle{288}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{36}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{18}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{72}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$ $$\displaystyle{144}\pi\frac{{m}^{{{2}}}}{{\sec{}}}$$

Derivatives
ANSWERED ### Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. $$\displaystyle{g{{\left({t}\right)}}}={\frac{{{1}}}{{{t}^{{{2}}}}}};{g}'{\left(-{1}\right)},{g}'{\left({2}\right)},{g}'{\left(\sqrt{{{3}}}\right)}$$

Derivatives
ANSWERED ### Using the definition, calculate the derivatives of the functions. Then find the values of the derivatives as specified. $$\displaystyle{k}{\left({z}\right)}={\frac{{{1}-{z}}}{{{2}{z}}}};{k}'{\left(-{1}\right)},{k}'{\left({1}\right)},{k}'{\left(\sqrt{{{2}}}\right)}$$

Rational functions
ANSWERED ### Graph the rational functions. $$y=\frac{(x^{3})+x-2)}{(x-(x^{2})}$$

Rational functions
ANSWERED ### determine whether the statement is true or false. Justify your answer. When your attempt to find the limit of a rational function yields the indeterminate form 0/0, the rational function’s numerator and denominator have a common factor.

Polynomial graphs
ANSWERED ### For the following exercise, for each polynomial, a. find the degree. b. find the zeros, if any. c. find the y-intercept(s), if any. d. use the leading coefficient to determine the graph’s end behavior. and e. determine algebraically whether the polynomial is even, odd, or neither. $$\displaystyle{f{{\left({x}\right)}}}=-{3}{x}^{{2}}+{6}{x}$$

Polynomial graphs
ANSWERED ### For the following exercise, for each polynomial, a. find the degree. b. find the zeros, if any. c. find the y-intercept(s), if any. d. use the leading coefficient to determine the graph’s end behavior. and e. determine algebraically whether the polynomial is even, odd, or neither. $$\displaystyle{f{{\left({x}\right)}}}={3}{x}−{x}^{{3}}$$

Transformations of functions
ANSWERED ### State the parent function that must be transformed to create the graph of each of the following functions. Then describe the transformations that must be applied to the parent function. a) $$\displaystyle{y}={\left(\frac{{5}}{{4}}\right)}{x}^{{4}}+{3}$$ b) $$y=3x-4$$ c) $$\displaystyle{y}={\left({3}{x}+{4}\right)}^{{3}}-{7}$$ d) $$\displaystyle{y}=-{\left({x}+{8}\right)}^{{4}}$$ e) $$y=-4.8(x-3)(x-3)$$ f) $$\displaystyle{2}{\left({\left(\frac{{1}}{{5}}\right)}{x}+{7}\right)}^{{3}}-{4}$$

Transformations of functions
ANSWERED ### Find the limit (if it exists) and discuss the continuity of the function. $$\lim_{(x,y,z) \rightarrow (-3,1,2)}\frac{\ln z}{xy-z}$$

Calculus and Analysis
ANSWERED ### Find derivatives for the functions. Assume a, b, c, and k are constants. $$\displaystyle{y}'={\left({17}{x}+{24}{x}^{{{\frac{{{1}}}{{{2}}}}}}\right)}'$$

Exponential models
ANSWERED ### Sociologists have found that information spreads among a population at an exponential rate. Suppose that the function $$y=525(1−e^{−0.038}t)$$ models the number of people in a town of 525 people who have heard news within t hours of its distribution. How many people will have heard about the opening of a new grocery store within 24 hours of the announcement?

Rational functions
ANSWERED ### Graph the rational functions. $$\displaystyle{y}=\frac{x}{{{\left({x}^{2}\right)}-{2}}}$$

Modeling
ANSWERED ### The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day 1 and assuming $$α=0.1$$ and a carrying capacity of 10 million cells, how long does it take to reach “detection” stage at 5 million cells?
ANSWERED 