If f(x) = -πx³, determine f'(x) from the first principles. Determine dy/dx if y = (1 − √x)³. Determine g'(x)…

Question

If f(x) = -πx³, determine f'(x) from the first principles. Determine dy/dx if y = (1 − √x)³. Determine g'(x) if g(x) = x(x² − 100) − x² + 100 / x − 1. If h(x) = 7 − 2/x², calculate the value of h'(2) + 4.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to Questions 7 and 8. Question 7 7.1 Determine f'(x) from the first principles if f(x) = −π x². Step 1: Write down the formula for differentiation from first principles. f'(x) = lim_(h → 0) (f(x+h) − f(x))/(h); Step 2: Substitute f(x) = −π x² and f(x+h) = −π (x+h)² into the formula. f'(x) = lim_(h → 0) (−π (x+h)² − (−π x²))/(h); f'(x) = lim_(h → 0) (−π (x² + 2xh + h²) + π x²)/(h); f'(x) = lim_(h → 0) (−π x² − 2π xh − π h² + π x²)/(h); f'(x) = lim_(h → 0) (−2π xh − π h²)/(h) Step 3: Factor out h from the numerator and simplify. f'(x) = lim_(h → 0) (h(−2π x − π h))/(h); f'(x) = lim_(h → 0) (−2π x − π h) Step 4: Evaluate the limit as h → 0. f'(x) = −2π x − π (0); f'(x) = −2π x; f'(x) = −2π x; 7.2 Determine dy/dx if y = (1−√(x))². Step 1: Expand the expression for y. y = (1 − x^(1/2))²; y = 1 − 2x^(1/2) + (x^(1/2))²; y = 1 − 2x^(1/2) + x Step 2: Differentiate each term with respect to x. dy/dx = d/dx(1) − d/dx(2x^(1/2)) + d/dx(x); dy/dx = 0 − 2 (1/2x^(1/2−1)) + 1; dy/dx = −x^(−1/2) + 1; dy/dx = −(1)/(√(x)) + 1; dy/dx = 1 − (1)/(√(x)); 7.3 Determine g'(x) if g(x) = (x(x²−100)−x²+100)/(x−1). Step 1: Simplify the expression for g(x) by factoring the numerator. g(x) = (x(x²−100) − (x²−100))/(x−1); g(x) = ((x−1)(x²−100))/(x−1) For x ≠ 1, we can cancel the (x−1) terms: g(x) = x² − 100; Step 2: Differentiate g(x) = x² − 100 with respect to x. g'(x) = d/dx(x²) − d/dx(100); g'(x) = 2x − 0; g'(x) = 2x; g'(x) = 2x; 7.4 If h(x) = 7 − (2)/(x²), calculate the value of h'(2)+4. Step 1: Rewrite h(x) using negative exponents. h(x) = 7 − 2x⁻² Step 2: Determine h'(x). h'(x) = d/dx(7) − d/dx(2x⁻²); h'(x) = 0 − 2(−2x⁻²⁻¹); h'(x) = 4x⁻³; h'(x) = (4)/(x³) Step 3: Calculate h'(2). h'(2) = (4)/((2)³); h'(2) = 4/8; h'(2) = 1/2 Step 4: Calculate h'(2)+4. h'(2)+4 = 1/2 + 4; h'(2)+4 = 1/2 + 8/2; h'(2)+4 = 9/2; h'(2)+4 = 9/2 Question 8 The graph below represents the function f(x) = −x³ + dx + q with A(−1;0) and B(2;0). 8.1 Show that d=3 and q=2. Step 1: Use point A(−1;0) to form an equation. Substitute x=−1 and f(x)=0 into f(x) = −x³ + dx + q: 0 = −(−1)³ + d(−1) + q; 0 = −(−1) − d + q; 0 = 1 − d + q; d − q = 1 (Equation 1) Step 2: Use point B(2;0) to form another equation. Substitute x=2 and f(x)=0 into f(x) = −x³ + dx + q: 0 = −(2)³ + d(2) + q; 0 = −8 + 2d + q; 2d + q = 8 (Equation 2) Step 3: Solve the system of equations. Add Equation 1 and Equation 2: (d − q) + (2d + q) = 1 + 8; 3d = 9; d = 3; Step 4: Substitute d=3 into Equation 1 to find q. 3 − q = 1; −q = 1 − 3; −q = −2; q = 2; Therefore, d=3 and q=2. d=3 and q=2 8.2 Determine the coordinates of turning point C. Step 1: Write the full equation for f(x) using d=3 and q=2. f(x) = −x³ + 3x + 2 Step 2: Find the derivative f'(x) to locate turning points. f'(x) = d/dx(−x³ + 3x + 2); f'(x) = −3x² + 3; Step 3: Set f'(x)=0 to find the x-coordinates of the turning points. −3x² + 3 = 0; −3x² = −3; x² = 1; x = ± 1; The turning points occur at x=−1 and x=1. From the graph, point C is the turning point with a positive x-coordinate, so x_C = 1. Step 4: Substitute x_C=1 into f(x) to find the y-coordinate of C. f(1) = −(1)³ + 3(1) + 2; f(1) = −1 + 3 + 2; f(1) = 4 The coordinates of turning point C are (1;4). C(1;4)

If f(x) = -πx³, determine f'(x) from the first principles. Determine dy/dx if y = (1 − √x)³. Determine g'(x) if g(x) = x(x² - 100) - x² + 100 / x - 1. If h(x) = 7 − 2/x², calculate the value of h'(2) + 4.

Answers

Step-by-step solution

Part 7

Part 7.1· 4 steps
$f'(x) = -2\pi x$

Write down the formula for differentiation from first principles.

Substitute $f(x) = -\pi x^2$ and $f(x+h) = -\pi (x+h)^2$ into the formula.

Factor out $h$ from the numerator and simplify.

Evaluate the limit as $h \to 0$ .

Part 7.2· 2 steps
$\frac{dy}{dx} = 1 - \frac{1}{\sqrt{x}}$

Expand the expression for $y$ .

Differentiate each term with respect to $x$ .

Part 7.3· 2 steps
$g'(x) = 2x$

Simplify the expression for $g(x)$ by factoring the numerator.

Differentiate $g(x) = x^2 - 100$ with respect to $x$ .

Part 7.4· 4 steps
$h'(2)+4 = \frac{9}{2}$

Rewrite $h(x)$ using negative exponents.

Determine $h'(x)$ .

Calculate $h'(2)$ .

Calculate $h'(2)+4$ .

Part 8

Part 8.1· 4 steps
$d=3$ and $q=2$

Use point $A(-1;0)$ to form an equation.

Use point $B(2;0)$ to form another equation.

Solve the system of equations.

Substitute $d=3$ into Equation 1 to find $q$ .

Part 8.2· 4 stepsC(1;4)

Write the full equation for $f(x)$ using $d=3$ and $q=2$ .

Find the derivative $f'(x)$ to locate turning points.

Set $f'(x)=0$ to find the x-coordinates of the turning points.

Substitute $x_C=1$ into $f(x)$ to find the y-coordinate of C.

Try one yourself

Still stuck on this one?

More Mathematics Questions

If f(x) = -πx³, determine f'(x) from the first principles. Determine dy/dx if y = (1 − √x)³. Determine g'(x) if g(x) = x(x² - 100) - x² + 100 / x - 1. If h(x) = 7 − 2/x², calculate the value of h'(2) + 4.

Answers

Step-by-step solution

Part 7

Part 7.1· 4 stepsf′(x)=−2πxf'(x) = -2\pi xf′(x)=−2πx

Write down the formula for differentiation from first principles.

Substitute f(x)=−πx2f(x) = -\pi x^2f(x)=−πx2 and f(x+h)=−π(x+h)2f(x+h) = -\pi (x+h)^2f(x+h)=−π(x+h)2 into the formula.

Factor out hhh from the numerator and simplify.

Evaluate the limit as h→0h \to 0h→0.

Part 7.2· 2 stepsdydx=1−1x\frac{dy}{dx} = 1 - \frac{1}{\sqrt{x}}dxdy​=1−x​1​

Expand the expression for yyy.

Differentiate each term with respect to xxx.

Part 7.3· 2 stepsg′(x)=2xg'(x) = 2xg′(x)=2x

Simplify the expression for g(x)g(x)g(x) by factoring the numerator.

Differentiate g(x)=x2−100g(x) = x^2 - 100g(x)=x2−100 with respect to xxx.

Part 7.4· 4 stepsh′(2)+4=92h'(2)+4 = \frac{9}{2}h′(2)+4=29​

Rewrite h(x)h(x)h(x) using negative exponents.

Determine h′(x)h'(x)h′(x).

Calculate h′(2)h'(2)h′(2).

Calculate h′(2)+4h'(2)+4h′(2)+4.