∫ calculus

1. The problem is to identify which graphs represent the gradient (derivative) functions of given graphs A, B, and D, and to find one other pair of graphs where one is the gradient

1. **Problem:** Find the inverse of the logarithmic function $$y = \log_5(2x - 1) - 7$$. 2. **Recall the definition of inverse functions:** If $$y = f(x)$$, then the inverse functi

1. **Problem:** Sketch and analyze the function $f$ with the given concavity and monotonicity properties. 2. **Key concepts:**

1. The problem is to understand if you need to find the derivative every time to calculate the rate of change. 2. The rate of change of a function at a point is given by the deriva

1. **State the problem:** We are given the temperature function $$T = 5 \sin\left(\frac{\pi}{12} x\right) + 23$$ where $x$ is the number of hours after sunrise. We need to find the

1. The problem is to find the derivative of the function $y = x^{\cos^{-1} x}$. 2. We use the formula for the derivative of $y = f(x)^{g(x)}$:

1. **Problem Statement:** Find the second derivative of the function $$f(\theta) = \frac{1}{3 + 2\cos\theta}$$ with respect to $$\theta$$. 2. **Recall the formula:** To find the se

1. The problem asks for the value of the limit $$\lim_{x \to -4^+} f(x)$$, which means the limit of the function $f(x)$ as $x$ approaches $-4$ from the right side. 2. From the grap

1. **State the problem:** Find the limit $$\lim_{x \to 0} (1 + 2^x)^{\frac{3}{x}}.$$\n\n2. **Recall the formula:** Limits of the form $$\lim_{x \to 0} (1 + f(x))^{\frac{1}{x}} = e^

1. Problem: Find the derivative of $y = \sin \alpha x$. 2. Formula: The derivative of $\sin u$ with respect to $x$ is $\cos u \cdot \frac{du}{dx}$.

1. Problem: Find the derivative of the function $f(x) = \frac{2}{3} x^{-3}$.\n\n2. Recall the power rule for derivatives: If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$.\n\n3. Apply

1. Problem: Find the derivative of the function $f(x) = -5 \sqrt[3]{x^2}$. 2. Rewrite the function using exponents: $f(x) = -5 x^{\frac{2}{3}}$.

1. We are given the function $$f(t) = 5t^2 - \frac{1}{t^3} + 3t - \sqrt{t} + 1$$ and asked to find the second derivative $$f''(t)$$ at $$t=1$$. 2. First, recall the rules for deriv

1. **State the problem:** Find the definite integral $$\int_0^1 \frac{t^2 + 1}{t^4 + 1} \, dt.$$\n\n2. **Recall the formula and approach:** We want to integrate a rational function

1. The problem asks for the value of $f(5)$ and the limits of $f(x)$ as $x$ approaches 5 from the left, right, and both sides. 2. From the graph description, near $x=5$, the bottom

1. **State the problem:** We want to evaluate the integral $$\int_0^{1} \frac{x^x (x^{2x} + 1) (\ln x + 1)}{x^{4x} + 1} \, dx.$$\n\n2. **Analyze the integrand:** The integrand is $

1. **State the problem:** Find the second derivative $f''(t)$ of the function $$f(t) = 5t^2 - \frac{1}{t^3} + 3t - \sqrt{t} + 1$$ and then evaluate it at $t=1$. 2. **Rewrite the fu

1. **Problem (a):** Find the gradient of the curve $y = 3x^4 - 2x^2 + 5x - 2$ at points $(0, -2)$ and $(1, 4)$. The gradient of a curve at any point is given by the derivative $\fr

1. We are asked to find the limit: $$\lim_{x \to +\infty} \left(1 - \frac{3}{x}\right)^x$$ 2. This is a classic limit of the form $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)

1. **Problem statement:** Find the limit $$\lim_{x \to +\infty} \left(1 - \frac{3}{x}\right)^x$$. 2. **Recall the formula:** The limit $$\lim_{x \to \infty} \left(1 + \frac{a}{x}\r

1. The problem is to find the derivative of the function $f(x) = x^2$. 2. The formula for the derivative of a power function $f(x) = x^n$ is given by the power rule: $$\frac{d}{dx}